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Radicals/212144: sqrt(2x+5)=x-5
1 solutions
Answer 160220 by drj(1380) About Me  on 2009-09-09 11:54:57 (Show Source):
You can put this solution on YOUR website!
Solve for x in the following equation

sqrt%282x%2B5%29=x-5

Step 1. Square Both Sides of Equation

%28sqrt%282x%2B5%29%29%5E2=%28x-5%29%5E2

2x%2B5=x%5E2-10x%2B25

Step 2. Simplify to get a quadratic equation. Subtract 2x+5 from both sides of equation

2x%2B5-%282x%2B5%29=%28x-5%29%5E2=x%5E2-10x%2B25-%282x%2B5%29

0=x%5E2-10x%2B25-%282x%2B5%29

0=x%5E2-8x%2B20

Step 3. Now use the quadratic equation

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=1, b=-8 and c=20 and follow steps below:

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-2x%2B-3+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-2%29%5E2-4%2A1%2A-3=16.

Discriminant d=16 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+16+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+16+%29%29%2F2%5C1+=+3
x%5B2%5D+=+%28-%28-2%29-sqrt%28+16+%29%29%2F2%5C1+=+-1

Quadratic expression 1x%5E2%2B-2x%2B-3 can be factored:
1x%5E2%2B-2x%2B-3+=+1%28x-3%29%2A%28x--1%29
Again, the answer is: 3, -1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B-3+%29



Step 4. x=3 and x=-1 are solutions to the equation.

For Step-By-Step Videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.

Human-and-algebraic-language/212131: This question is from textbook Elementary and Intermediate Algebra
Tickets for a concert were sold to adults for
$3 and to students for $2. If the total receipts were $824
and twice as many adult tickets as student tickets were
sold, then how many of each were sold?
1 solutions
Answer 160219 by drj(1380) About Me  on 2009-09-09 11:42:52 (Show Source):
You can put this solution on YOUR website!
Tickets for a concert were sold to adults for
$3 and to students for $2. If the total receipts were $824
and twice as many adult tickets as student tickets were
sold, then how many of each were sold?

Step 1. Let x = number of tickets sold to adults

Step 2. Let y = number of tickets sold to students

Step 3. Total receipts = 824

3x%2B2y=824

Step 4. x=2y since twice as many adults tickets sold when compared to student tickets. y must be a smaller value than x.

Step 5. Substitute x of Step 4 into Step 3.
3x%2B2y=824
3%282y%29%2B2y=824
6y%2B2y=824
8y=824
y=824%2F8
y=103

Step 6. 103 Student tickets were sold and 206 adult tickets were sold where x=2y=2*103=206.

For Step-By-Step Videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.

Equations/212136: i need a solution to the ordinary differential equation
x"+2x'+2x=0
1 solutions
Answer 160217 by drj(1380) About Me  on 2009-09-09 11:27:11 (Show Source):
You can put this solution on YOUR website!
I need a solution to the ordinary differential equation
x"+2x'+2x=0

Step 1. Assume the solution is

x=e%5Emx

If this is true, then we need to solve for m.

Step 2. Take the derivatives of x:

x" = m%5E2e%5Emx

x' = me%5Emx+

Step 3. Substitute above derivatives and x into the given equation

x"+2x'+2x= 0=m%5E2e%5Emx%2B2me%5Emx%2B2e%5Emx

Step 4. Factor out

e%5Emx

to get

%28e%5Emx%29%28m%5E2%2B2m%2B2%29=0

Step 5. The only way to get 0 is that the quadratic expression is zero. That is,

m%5E2%2B2m%2B2=0

Now, we can solve for m using the quadratic formula below.

m+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where

a=1, b=2 and c=2

Step 6. See standard procedure of solving quadratic equation below:

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation am%5E2%2Bbm%2Bc=0 (in our case 1m%5E2%2B2m%2B2+=+0) has the following solutons:

m%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%282%29%5E2-4%2A1%2A2=-4.

The discriminant -4 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.


In the field of imaginary numbers, the square root of -4 is + or - sqrt%28+4%29+=+2.

The solution is m%5B12%5D+=+%28-2%2B-+i%2Asqrt%28+-4+%29%29%2F2%5C1+=++%28-2%2B-+i%2A2%29%2F2%5C1+

Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B2+%29




Step 7. Note the roots are complex. So your solution will consists of complex exponentials. Also, please ignore the graph since it's only applicable for real roots.


Using the above roots m1 and m2, the solution is

y=c1e%5Em1%2Bc2e%5Em2

where the c1 and c2 are arbitrary constants. We need initial values to get values of c1 and c2.


I believe the above problem is above the skill level of algebra. However,
for Step-By-Step Videos on Differential Equations, Circuit Analysis and Design,please visit http://wwww.FreedomUniversity.TV

Triangles/212135: the hypotenuse of a right triangle is twice as long as one of the legs and 6 inches longer than the other. what are the lengths of the sides of the triangle
1 solutions
Answer 160213 by drj(1380) About Me  on 2009-09-09 11:06:00 (Show Source):
You can put this solution on YOUR website!
The hypotenuse of a right triangle is twice as long as one of the legs and 6 inches longer than the other. what are the lengths of the sides of the triangle

Step 1. Let x be length of one of the legs. Then x+6 is the length of the other leg. And length of hypotenuse is 2x.

Step 2. Pythagorean Theorem. Sum of the squares of the legs is equal to the square of the hypotenuse.

x%5E2%2B%28x%2B6%29%5E2=%282x%29%5E2

Simplify right side,

x%5E2%2B%28x%2B6%29%5E2=4x%5E2

Subtract x^2 from both sides

x%5E2%2B%28x%2B6%29%5E2-x%5E2=4x%5E2-x%5E2

x%5E2%2B%28x%2B6%29%5E2-x%5E2=3x%5E2

Simplify left side and multiply out the square term of x+6

x%5E2%2B12x%2B36=3x%5E2

Subtract the left side of equation on both sides by itself to get zero on the left side. That is

x%5E2%2B12x%2B36-%28x%5E2%2B12x%2B36%29=3x%5E2-%28x%5E2%2B12x%2B36%29

Simplifying both sides of equation yields
0=2x%5E2-12x-36

Step 3. This is now a quadratic equation where we can use the quadratic formula.
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
where a=2, b=-12, c=-36

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-12x%2B-36+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-12%29%5E2-4%2A2%2A-36=432.

Discriminant d=432 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--12%2B-sqrt%28+432+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-12%29%2Bsqrt%28+432+%29%29%2F2%5C2+=+8.19615242270663
x%5B2%5D+=+%28-%28-12%29-sqrt%28+432+%29%29%2F2%5C2+=+-2.19615242270663

Quadratic expression 2x%5E2%2B-12x%2B-36 can be factored:
2x%5E2%2B-12x%2B-36+=+2%28x-8.19615242270663%29%2A%28x--2.19615242270663%29
Again, the answer is: 8.19615242270663, -2.19615242270663. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-12%2Ax%2B-36+%29




Step 4. Pick the positive number since we have only positive lengths. In this case, it's 8.196. so
x=8.196 Length of one leg

x%2B6=14.196 Length of other Leg

2x=28.392 Length of hypotenuse.

You can use the Pythagorean Theorem to check if its close to verify your answer.

For Step-By-Step Videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.

For more info, contact Professor John Santiago at john@e-liteworks.com

Miscellaneous_Word_Problems/212102: Find the x intercepts for the parabola y = x2 – 6x + 5.
Find the vertex of the parabola y = - 2x2 + 8x + 4


1 solutions
Answer 160210 by drj(1380) About Me  on 2009-09-09 10:18:19 (Show Source):
You can put this solution on YOUR website!
Find the x intercepts for the parabola y = x2 – 6x + 5.
Step 1. For this problem y=0 since we want x-intercepts.

Step 2. Use quadratic formula:
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
where a=1, b=-6 and c= 5
Step 3. The following steps shows how to solve the above equation in Step 2 where you will find x=1 and x=5. Note the parabola intercepts x-axis when y=6 at these points.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-6x%2B5+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-6%29%5E2-4%2A1%2A5=16.

Discriminant d=16 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--6%2B-sqrt%28+16+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+16+%29%29%2F2%5C1+=+5
x%5B2%5D+=+%28-%28-6%29-sqrt%28+16+%29%29%2F2%5C1+=+1

Quadratic expression 1x%5E2%2B-6x%2B5 can be factored:
1x%5E2%2B-6x%2B5+=+1%28x-5%29%2A%28x-1%29
Again, the answer is: 5, 1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-6%2Ax%2B5+%29





Problem: Find the vertex of the parabola y = - 2x2 + 8x + 4

+-2x%5E2%2B8x%2B4

Step 1. Graph is shown below and the vertex is at (2,12)
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case -2x%5E2%2B8x%2B4+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%288%29%5E2-4%2A-2%2A4=96.

Discriminant d=96 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-8%2B-sqrt%28+96+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%288%29%2Bsqrt%28+96+%29%29%2F2%5C-2+=+-0.449489742783178
x%5B2%5D+=+%28-%288%29-sqrt%28+96+%29%29%2F2%5C-2+=+4.44948974278318

Quadratic expression -2x%5E2%2B8x%2B4 can be factored:
-2x%5E2%2B8x%2B4+=+-2%28x--0.449489742783178%29%2A%28x-4.44948974278318%29
Again, the answer is: -0.449489742783178, 4.44948974278318. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-2%2Ax%5E2%2B8%2Ax%2B4+%29



For Step-By-Step Videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.


Travel_Word_Problems/212115: Bob is making a 40-kilometer boat trip. if he travels at 30 kilometers per hour for the first 10 kilometers, and then at 15 kilometers per hour for the rest of the trip, how many minutes more will it take him than if he tarvels the entire trip at 20 kilometers per hour?
1 solutions
Answer 160202 by drj(1380) About Me  on 2009-09-09 09:53:39 (Show Source):
You can put this solution on YOUR website!
Bob is making a 40-kilometer boat trip. If he travels at 30 kilometers per hour for the first 10 kilometers, and then at 15 kilometers per hour for the rest of the trip, how many minutes more will it take him than if he travels the entire trip at 20 kilometers per hour?

Step 1. Translate following into an equation: He travels at 30 kilometers per hour for the first 10 kilometers. We note that

Distance=Velocity*Time

10+=+30x
x=10%2F30 hours
x=10%2F30%2A60 minutes where 60 minutes = 1 hours
x=20 minutes


where Distance = 10 km, x is the time he travels at 30 kilometers per hours and 30x is the distance traveled.

Step 2. 15 kilometers per hour for the rest of the trip which in this case is 10 kilometers = 40-30.
10=15y
y=2%2F3 hours
y=%282%2F3%29%2A60 minutes
y=40 minutes

Step 3. Add Step 1 and Step 2 for Total Time.

Total time = 20 minutes + 40 minutes = 60 minutes = 1 hour

Step 4. Last part of the problem means he travels 40 km at 20 kilometers per hour.

40+=+20z
z+=+40%2F20 hour
z+=+1%2F2+ hour
+z+=+1%2F2%2860%29 minutes
+z+=+30 minutes

where z is the amount of time for this part of the problem.

Step 5. The amount of time in Step 3 is 60 minutes and amount of time in Step 4 is 30 minutes. So Step 3 is 30 minutes more than Step 4.

For more Step-By-Step Videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.










Miscellaneous_Word_Problems/212105: $1500 is deposited every year in an account yielding 6% interest compounded annually, how much money will have been saved after 10 years?
1 solutions
Answer 160195 by drj(1380) About Me  on 2009-09-09 09:15:54 (Show Source):
You can put this solution on YOUR website!
$1500 is deposited every year in an account yielding 6% interest compounded annually, how much money will have been saved after 10 years?

Step 1. Money after First year = 1500*1.06 (Money after first year is bigger than initial investment. That's why 1.06 must be bigger than 1 where 6% is 0.06. Then add 1 to get 1.06)

Step 2. Money after Second year = First Year*1.06= 1500*1.06*1.06

Step 3. Money after Third year = Second Year*1.06= 1500*1.06*1.06*1.06

Step 4. Base on Steps 1-3, there is a pattern. So after n years then the general formula is

Pn=1500%2A1.06%2A10%5En

where n is the number of years and Pn is the amount of money after n years. In this case n=10 and payment after ten year is labelled as P10

Step 5. Solve equation in Step 4.
P10=1500%2A%281.06%5E10%29
now 1.06%5E10=1.790847697
So
P10=1500%2A1.790847697
P10=2686.27

Step 6. So at the end of 10 years an initial investment of $1500 compounded annually at 6% is $2686.27. You almost doubled your money. You should double your money at the end of 12 years. You can check by substituting n=12.

Happy Investing! Dr J

For Step-By-Step Videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.


For more questions in investing calculations, please contact Dr J at john@e-liteworks.com.


Human-and-algebraic-language/212113: YOU PAY A TOLL IN EITHER ALL QUARTERS OR ALL DIMES. THE TOLL IS 3 MORE DIMES THAN QUARTERS. WHAT IS THE TOLL?
1 solutions
Answer 160192 by drj(1380) About Me  on 2009-09-09 08:57:47 (Show Source):
You can put this solution on YOUR website!
YOU PAY A TOLL IN EITHER ALL QUARTERS OR ALL DIMES. THE TOLL IS 3 MORE DIMES THAN QUARTERS. WHAT IS THE TOLL?

Step 1. Total cost = 10x = 25y


where x is the number of dimes
and y is the number of quarters

Step 2. y=x-3 since toll is 3 more dimes than quarters. That is, number of quarters is less than number of times,

Step 3. Substitute y of Step 2 into Step 1 .
10x=25y
10x=25%28x-3%29
10x=25x-75
Step 4. Subtract 10x from both sides of equation to make left side = 0
10x-10x=25x-75-10x
0=15x-75
Step 3. Now add 75 to both sides of equation to isolate x on one side.
0%2B75=15x-75%2B75
15x=75
Step 6. Isolate x by dividing 15 to both sides of equation
15x%2F15=75%2F15=5
Step 7. x=5 (x is the number of dimes) so the cost of the toll is

Total Cost = 10%2A5=50+cents

As a check with y as the number of quarters then
y=5-3=2,
so
Total Cost = 25%2A2=50+cents+
So the answer also checks out with the number of quarters.

For Step-By-Step Videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.

Contact Dr J for more info at john@e-liteworks.com


Miscellaneous_Word_Problems/212107: Debbie plans on investing enough money so that she will have $2000 at the end of two years. How much money will she have to invest now if her money will grow according to the following formula: 2000 = P (1.025)^4?
1 solutions
Answer 160187 by drj(1380) About Me  on 2009-09-09 08:27:02 (Show Source):
You can put this solution on YOUR website!
Debbie plans on investing enough money so that she will have $2000 at the end of two years. How much money will she have to invest now if her money will grow according to the following formula: 2000 = P (1.025)^4?

Step 1. Solve for P where P is the initial investment. Interest in this case is 2.5 % compounded semiannually since in this case it's compounded 4 times in 2 years
Divide by 1.025%5E4=+1.025%2A1.025%2A1.025%2A1.025=1.103812891 to both sides of equation to isolate P.
2000%2F1.103812891=P%281.025%5E4%29%2F1.025%5E4=P
Step 2. P=$1811.90

Miscellaneous_Word_Problems/212106: Richard drove his rig at ‘x’ mph for 3 hours, then increased his speed to (x + 15) mph and drove 2 more hours. Write a polynomial that represents the total distance that he traveled. How much distance he covered if x = 45 Mph?
1 solutions
Answer 160185 by drj(1380) About Me  on 2009-09-09 08:13:09 (Show Source):
You can put this solution on YOUR website!
Richard drove his rig at ‘x’ mph for 3 hours, then increased his speed to (x + 15) mph and drove 2 more hours. Write a polynomial that represents the total distance that he traveled. How much distance he covered if x = 45 Mph?

Step 1. Distance=Velocity * Time

Total Distance in miles = 3x%2B2%28x%2B15%29+=+3x%2B2x%2B30+=+5x+%2B+30+

where

3x is the distance in miles traveling for 3 hours at x mph
2%28x%2B15%29=2x%2B30 is the distance in miles traveling for 2 hours at x+15 mph

Step 2. Substitute x=45 Mph into Total Distance equation of Step 1

+5x%2B30+=+5+%2845%29%2B30+=+255+

Step 3. Total Distance traveled is 255 miles.

For Step-By-Step Videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.




Average/212108: what is the average of 35.74
1 solutions
Answer 160184 by drj(1380) About Me  on 2009-09-09 07:59:48 (Show Source):
You can put this solution on YOUR website!
I think what you meant is what is the average of 35 and 74?

Step 1. Add 35 and 74.
35%2B74=109

Step 2. Count the number of values. In this case it's 2.

Step 3. Divide Total of 109 by 2.

109%2F2=49.5

For Step-By-Step Videos in Introduction to Algebra, visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.

Travel_Word_Problems/212069: Russ and Janet are running. Russ runs at 7mph, Janet at 5mph. If the run has a staggered start, and Janet starts first with Russ starting 10 minutes later, how long before he catches up with her?
1 solutions
Answer 160157 by drj(1380) About Me  on 2009-09-08 23:47:05 (Show Source):
You can put this solution on YOUR website!
Russ and Janet are running. Russ runs at 7mph, Janet at 5mph. If the run has a staggered start, and Janet starts first with Russ starting 10 minutes later, how long before he catches up with her?

Step 1. Find distance Janet travels just before Russ starts running.

(5 miles/hour) * (1 hour/60 minutes) * (10 minutes) = 5%2A10%2F60=5%2F6 miles

Step 2. You need to realize that Russ and Janet must travel the same distance when Russ catches up.

distance = velocity * time

Step 3. Russ traveled 7%2Ax where x is the time (in hours) he traveled

Step 4. Janet traveled 5%2F6%2B5%2Ax where same x is the time she traveled
Step 5. Set equations equal: Step 3 = Step 4
7%2Ax=5%2F6%2B5%2Ax
Step 6. Subtract 5x from both sides of equation in previous step.

7x-5x=5%2F6%2B5x-5x
2x=5%2F6
Step 7. Divide by 2 to solve for x

2x%2F2=5%2F%282%2A6%29

Step 8. Answer is
x=5%2F12 hour or x=5%2F12%2A60=25 minutes where 60 minutes = 1 hour


For Step-By-Step Videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.

Functions/212060: x+15/6-2x+8/2=-4
1 solutions
Answer 160153 by drj(1380) About Me  on 2009-09-08 23:28:12 (Show Source):
You can put this solution on YOUR website!
x+15/6-2x+8/2=-4

Step 1. To solve for x collect like terms on left side. Simplify 8/2=4.

%28x-2x%29%2B15%2F6%2B4=-4
Step 2. Subtract 15%2F6%2B4 to both sides of equation to isolate x by itself on the left side
-x%2B15%2F6%2B4-15%2F16-4=-4-4-15%2F6 (left side is now -x when canceling terms)
-x=-4-4-15%2F6 (simplify left side)
-x=-8-15%2F6 (simplify right side -4-4=-8)
-x=-8%2A6%2F6-15%2F6 (put in common denominator)
-x=-%2848-15%29%2F6 (multiply by -1 to both sides of equation toget final answer below)
x=-%2833%29%2F6

Equations/212059: I was givien the following problem:
Write the equation 2x+3y=9 in function form and find f(-1) and f(6)

1 solutions
Answer 160143 by drj(1380) About Me  on 2009-09-08 22:32:21 (Show Source):
You can put this solution on YOUR website!
I was givien the following problem:

Write the equation 2x+3y=9 in function form and find f(-1) and f(6)

Step 1. Solve for y, such that y=f(x) where f(x) is a function of x.

Step 2. Subtract 2x in both sides of equation

2x%2B3y-2x=9-2x

3y=9-2x

Step 3. Divide 3 in both sides of equation

3y%2F3=%289-2x%29%2F3

y=3-2x%2F3

y=f%28x%29=3-2x%2F3

Step 4. f(-1) implies x=-1

y=f%28-1%29=3-2%28-1%29%2F3=3%2B2%2F3
y=f%28-1%29=9%2F3%2B2%2F3=11%2F3

Step 5. f(6) implies x=6

y=f%286%29=3-2%286%29%2F3=3%2B2%2A2=7

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Miscellaneous_Word_Problems/212048: This question is from textbook
The height of a triangle is 5 centimeters longer than three times its base. Find the base of the triangle if its area is 6 square centimeters
1 solutions
Answer 160139 by drj(1380) About Me  on 2009-09-08 22:14:54 (Show Source):
You can put this solution on YOUR website!
The height of a triangle is 5 centimeters longer than three times its base. Find the base of the triangle if its area is 6 square centimeters

Step 1. Area of Triangle = A+=+xh%2F2+=6

where x=base, h=height and A=area of is one-half times base times height

Step 2. h=3x%2B5 (The height of a triangle is 5 centimeters longer than three times its base)

Step 3. Substitute height in Step 2 into equation in Step 1.

A=xh%2F2=x%283%2Bx5%29%2F2
%283x%5E2%2B5x%29%2F2=6

Step 4. Multiply 2 in both sides of last equation in Step 3 to get rid of denominator.

2%283x%5E2%2B5x%29%2F2=2%2A6
3x%5E2%2B5x=12
Simplify by subtracting 12 from both sides of equation yields

3x%5E2%2B5x-12=0

Step 5. The last equation of Step 4 is simply a quadratic equation. Then use the quadratic formula:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=3, b=5, and c=-12

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B5x%2B-12+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%285%29%5E2-4%2A3%2A-12=169.

Discriminant d=169 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-5%2B-sqrt%28+169+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%285%29%2Bsqrt%28+169+%29%29%2F2%5C3+=+1.33333333333333
x%5B2%5D+=+%28-%285%29-sqrt%28+169+%29%29%2F2%5C3+=+-3

Quadratic expression 3x%5E2%2B5x%2B-12 can be factored:
3x%5E2%2B5x%2B-12+=+3%28x-1.33333333333333%29%2A%28x--3%29
Again, the answer is: 1.33333333333333, -3. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B5%2Ax%2B-12+%29



Step 6. Pick the positive number in the quadratic equation. In this case, we have the base x= 1.3333=4/3.

height is then h=3x%2B5=3%284%2F3%29%2B5=9 centimeters
Area is then %281%2F2%29%284%2F3%29%289%29=6 square centimeters. This is the same answer to our given answer. So it check out.

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Quadratic_Equations/211995: a farmer wants to set up a pigpen using 40 feet of fence to enclose a rectangular area of 51 square feet. what are the dimensions of the pigpen?
1 solutions
Answer 160116 by drj(1380) About Me  on 2009-09-08 20:59:44 (Show Source):
You can put this solution on YOUR website!
A farmer wants to set up a pigpen using 40 feet of fence to enclose a rectangular area of 51 square feet. what are the dimensions of the pigpen?

1. Perimeter of a rectangle is 2x+2y=40 where x=one side of rectangle and y=the adjacent side of x. We can simplify this as x+y=20 where we divided 2 on both sides of the equation. That is,

%282x%2B2y%29%2F2=40%2F2=20

x%2By=20

y=20-x

2. Area=xy=51 where area of rectangle is height times width.

3. Substitute y in Step 1 into Step 2.

xy=x%2820-x%29=51 Multiplying the terms will yield: 20x-x%5E2=51

4. Put everything on the left side to the right. That is, add -20x+x^2 from both sides of the equation to make the left side equal to zero.

20x-x%5E2-20x%2Bx%5E2=+51%2Bx%5E2-20x

Simplifying will yield

0+=+51%2Bx%5E2-20x

Finally, the above equation is equivalent to

x%5E2-20x%2B51=0

4. Now this is just a Quadratic Equation so we can use the formula

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=1, b=-20 and c=51

5. Use the following steps to solve the quadratic equation.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-20x%2B51+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-20%29%5E2-4%2A1%2A51=196.

Discriminant d=196 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--20%2B-sqrt%28+196+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-20%29%2Bsqrt%28+196+%29%29%2F2%5C1+=+17
x%5B2%5D+=+%28-%28-20%29-sqrt%28+196+%29%29%2F2%5C1+=+3

Quadratic expression 1x%5E2%2B-20x%2B51 can be factored:
1x%5E2%2B-20x%2B51+=+1%28x-17%29%2A%28x-3%29
Again, the answer is: 17, 3. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-20%2Ax%2B51+%29




6. The rectangular sides are 3 and 17. As a check 3*17=51 (Area is 51 square feet) and Perimeter is 2*(3+17)=40 ft. So everything checks out.

For Step-by-Step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.Tv/courses/IntroAlgebra

Equations/211998: Tony has twice as much money as Alicia. She has $16 less than Ralph. Together they have $200. How much money does each have?
1 solutions
Answer 160110 by drj(1380) About Me  on 2009-09-08 20:25:27 (Show Source):
You can put this solution on YOUR website!
Tony has twice as much money as Alicia. She has $16 less than Ralph. Together they have $200. How much money does each have?

Step 1. Let x=amount of dollars that Alicia has.

Step 2. Tony has twice as much as Alicia or 2x dollars

Step 3. Ralph has x+16 dollars since Alicia has $16 less than Ralph.

Step 4. Since the total amount is $200 among Alicia, Tony and Ralph, add Steps 1-3. x+2x+x+16=200. This yields. 4x+16=200 or 4x=184 where I subtracted 16 from both sides of the equation.

Step 5. Solving x=184/4=46 dollars.

a. Alicia has 46 dollars
b. Tony has 46*2=92 dollars.
c Ralph has 46+16=62 dollars.
d. Total should be $200. So, 46+92+62=200. So it checks out.

For Step-By-Step Videos in Introductory Algebra and more word problems, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.



Linear_Equations_And_Systems_Word_Problems/211973: Mike bought some kilograms of apples and some kilograms of oranges and spent
$27.25. The apples cost $3.10 per kilogram and oranges cost $4.95 per kilogram.
If apples become $1.55 more expensive and oranges become $1.15 chaper he will
spend $30. How many ograms of apples and oranges did Mike buy?
1 solutions
Answer 160107 by drj(1380) About Me  on 2009-09-08 19:33:51 (Show Source):
You can put this solution on YOUR website!
Mike bought some kilograms of apples and some kilograms of oranges and spent
$27.25. The apples cost $3.10 per kilogram and oranges cost $4.95 per kilogram.
If apples become $1.55 more expensive and oranges become $1.15 cheaper he will
spend $30. How many kilograms of apples and oranges did Mike buy?

1. Let x=number of kilograms of apples and y=number of kilograms of oranges.

2. Cost of apples = x*$3.10 =3.10x

3. Cost of oranges= y*4.95=4.95y

4. Total cost = 27.25 = 3.10x+4.95y

5. Let's look at "If apples become $1.55 more expensive and oranges become $1.15 cheaper he will spend $30."

a. For apples 3.10+1.55=4.65

b. For oranges 4.95-1.15=3.80

c. New total cost = 30 = 4.65x+3.80y

d. Note we still have the same number of apples x and oranges y.

6. So now we have two equations and two unknowns

a. 27.25 = 3.10x+4.95y

b. 30 = 4.65x+3.80y

Need to solve for x and y. The steps below will show x=4 and y=3


Solved by pluggable solver: SOLVE linear system by SUBSTITUTION
Solve:
We'll use substitution. After moving 4.95*y to the right, we get:
3.1%2Ax+=+27.25+-+4.95%2Ay, or x+=+27.25%2F3.1+-+4.95%2Ay%2F3.1. Substitute that
into another equation:
4.65%2A%2827.25%2F3.1+-+4.95%2Ay%2F3.1%29+%2B+3.8%5Cy+=+30 and simplify: Error: 'Can't call method "invoke_solver" on an undefined value at /home/ichudov/project_locations/algebra.com/templates/Algebra/Solver/PerlSolver.pm line 115. Can't call method "invoke_solver" on an undefined value at /home/ichudov/project_locations/algebra.com/templates/Algebra/Solver/PerlSolver.pm line 115. '. Can't call method "invoke_solver" on an undefined value at /home/ichudov/project_locations/algebra.com/templates/Algebra/Solver/PerlSolver.pm line 115. Can't call method "invoke_solver" on an undefined value at /home/ichudov/project_locations/algebra.com/templates/Algebra/Solver/PerlSolver.pm line 115.
error_location: 101




For Step-By-Step Videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.


Rate-of-work-word-problems/211955: The question is:
Find two positive integers tha differ by 4 and whose product is 221. I know the answer is 13 and 17, but would like to know how to arrive at that algebraicly.
1 solutions
Answer 160105 by drj(1380) About Me  on 2009-09-08 19:06:12 (Show Source):
You can put this solution on YOUR website!
The question is:

Find two positive integers that differ by 4 and whose product is 221. I know the answer is 13 and 17, but would like to know how to arrive at that algebraically.

Step 1. Two Positive Integers that differ by 4. Say n is one positive integer. Since the two numbers differ by 4, then the other must can be either n+4 or n-4. We'll choose n+4 but you can use n-4 and get similar results.

Step 2. Product is 221. This means that n(n+4)=221.


Step 3. Multiply the equation is step 2. n%28n%2B4%29=n%5E2%2B4n=221. This will reduce to a quadratic equation given as


n%5E2%2B4n-221=0

where we subtracted 221 from both sides of the equation in the first equation of Step 3.

Step 4. Now follow the process of solving a quadratic equation

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
where a=1, b=4 and c=-221

The steps are shown below. The solutions to the quadratic equation equation below are 13 and -17 and will intercept the x-axis in the parabola such that y=0. However, please ignore the graph for the moment since solution exceeded the limits for the graph.

Step 4a. Since we want a positive numbers then choose n=13. Therefore n+4=17.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B4x%2B-221+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%284%29%5E2-4%2A1%2A-221=900.

Discriminant d=900 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-4%2B-sqrt%28+900+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%284%29%2Bsqrt%28+900+%29%29%2F2%5C1+=+13
x%5B2%5D+=+%28-%284%29-sqrt%28+900+%29%29%2F2%5C1+=+-17

Quadratic expression 1x%5E2%2B4x%2B-221 can be factored:
1x%5E2%2B4x%2B-221+=+1%28x-13%29%2A%28x--17%29
Again, the answer is: 13, -17. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B4%2Ax%2B-221+%29



For step-by-step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.Tv/courses/IntroAlgebra

Polynomials-and-rational-expressions/211947: This question is from textbook Mathematics1
How do you tell what degree a polynomial is?
1 solutions
Answer 160104 by drj(1380) About Me  on 2009-09-08 18:14:45 (Show Source):
You can put this solution on YOUR website!
Degree of polynomial is the term which has the highest exponent.

for example the polynomial : s%5E3%2B2s%5E2%2B5 has degree 3.

I also have free step-by-step videos in Introduction to Algebra at http://www.FreedomUniversity.TV/courses/IntroAlgebra. Spread the word to your friends.

Quadratic_Equations/211918: This question is from textbook Intoductery Algebra
This is from a chapter on solving a quadratic equation by compleating a square. Please help me solve.
%284%29%2F%28m%5E2-7%29=1
Thank you.
1 solutions
Answer 160097 by drj(1380) About Me  on 2009-09-08 17:14:26 (Show Source):
You can put this solution on YOUR website!
This is from a chapter on solving a quadratic equation by completing a square. Please help me solve.
%284%29%2F%28m%5E2-7%29=1
Step 1. For this case completing the square was not needed. Let's get rid of the denominator by multiplying the denominator to both sides of the equation.

The equation %284%29%2F%28m%5E2-7%29=1 OR %284%29%28m%5E2-7%29%2F%28m%5E2-7%29=1%2A%28m%5E2-7%29 which will now be

%284%29=%28m%5E2-7%29

Step 2. Now let's subtract -4 from both sides of the equation.

%284-4%29=0=%28m%5E2-7-4%29

0=%28m%5E2-11%29 (note: the answer is just the m=sqrt%2811%29) plus or minus sqrt(11)



Step 3. So this is now a quadratic equation where

a=1, b=0, and c=-11 where we will use the quadratic formula

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
Step 4. See the steps below to solve this.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B0x%2B-11+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%280%29%5E2-4%2A1%2A-11=44.

Discriminant d=44 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-0%2B-sqrt%28+44+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%280%29%2Bsqrt%28+44+%29%29%2F2%5C1+=+3.3166247903554
x%5B2%5D+=+%28-%280%29-sqrt%28+44+%29%29%2F2%5C1+=+-3.3166247903554

Quadratic expression 1x%5E2%2B0x%2B-11 can be factored:
1x%5E2%2B0x%2B-11+=+1%28x-3.3166247903554%29%2A%28x--3.3166247903554%29
Again, the answer is: 3.3166247903554, -3.3166247903554. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B0%2Ax%2B-11+%29



Note: Step-By-Step Videos in Pre-Algebra can be found at http://www.FreedomUniversity.Tv/courses/IntroAlgebra.

Linear-equations/211912: How do you figure out this problem: -5x(2x-1)=3(x+4)
1 solutions
Answer 160094 by drj(1380) About Me  on 2009-09-08 16:52:03 (Show Source):
You can put this solution on YOUR website!
How do you figure out this problem: -5x(2x-1)=3(x+4)

Step 1. Multiply everything out on both sides of the equation. After we mess around with the equation, we will get a quadratic equation. Here it goes:

-5x(2x-1)=3(x+4) will yield: -10x^2+5x=3x+12.

Step 2. Now let's put everything on the right side so that the left side is zero. I like a positive number for the x^2 term.

Step 2a. Add 10x^2 to both sides of equation in Step 1:

10x^2-10x^2+5x=10x^2+3x+12

Note: The 10x^2 terms cancel out on the left side.

Step 2b. Simplify equation in step 2a.

5x=10x^2+3x+12

Step 3. Subtract 5x both sides of equation in Step 2b to make left side =0.

5x-5x=10x^2+3x-12-5x

Note: Collect like terms and simplify 3x-5x=-2x

Step 3a. Simplify to yield our final equation

0=10x^2-2x-12 OR 10x^2-2x-12=0

Step 4. Now we have a quadratic equation in step 3a.

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=10, b=-2, and c=-12.


Step 5. Substitute a, b, and c in the formula. The steps are shown below:

Note when the quadratic equation is a parabola when the values of x are real numbers. See graph where it intersects the x-axis. This is when y=0, hence our quadratic equation.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 10x%5E2%2B-2x%2B-12+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-2%29%5E2-4%2A10%2A-12=484.

Discriminant d=484 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+484+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+484+%29%29%2F2%5C10+=+1.2
x%5B2%5D+=+%28-%28-2%29-sqrt%28+484+%29%29%2F2%5C10+=+-1

Quadratic expression 10x%5E2%2B-2x%2B-12 can be factored:
10x%5E2%2B-2x%2B-12+=+10%28x-1.2%29%2A%28x--1%29
Again, the answer is: 1.2, -1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+10%2Ax%5E2%2B-2%2Ax%2B-12+%29



For Step-By-Step Videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra


Quadratic_Equations/211828: This question is from textbook Introductory Algebra
Find the x-intercept.
y=x^2+4x-1
I am stuck on this..please help!
1 solutions
Answer 160037 by drj(1380) About Me  on 2009-09-08 12:24:02 (Show Source):
You can put this solution on YOUR website!
Find the x-intercept.
y=x^2+4x-1
I am stuck on this..please help!

Step 1. The x-intercept occurs when y=0. So let's set y=0=x^2+4x-1.

Step 2. Now use the quadratic formula given as:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=1 b=4 and c=-1

Step 3. Substitute above values in quadratic formula and you get the following. Note the values of x where y=0 and observe them on the graph.

Also, note a quadratic equation is basically a parabola if the value of x are real numbers.


Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B4x%2B-1+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%284%29%5E2-4%2A1%2A-1=20.

Discriminant d=20 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-4%2B-sqrt%28+20+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%284%29%2Bsqrt%28+20+%29%29%2F2%5C1+=+0.23606797749979
x%5B2%5D+=+%28-%284%29-sqrt%28+20+%29%29%2F2%5C1+=+-4.23606797749979

Quadratic expression 1x%5E2%2B4x%2B-1 can be factored:
1x%5E2%2B4x%2B-1+=+1%28x-0.23606797749979%29%2A%28x--4.23606797749979%29
Again, the answer is: 0.23606797749979, -4.23606797749979. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B4%2Ax%2B-1+%29



If you have any questions, I have a video also on the quadratic equation and you can contact Dr J at john@e-liteworks.com or visit my website at http://www.FreedomUniversity.TV for more step-by-step videos in math.

Polynomials-and-rational-expressions/211437: I need help factoring some equations please help:
2. 16xy + 32yz – 20xyz
3. 9x2 – 64
4. 5x3 – 125u2x
5. x2 + 7x + 12
Please explain to me how to do these because I am confused and old.
1 solutions
Answer 159812 by drj(1380) About Me  on 2009-09-07 11:38:56 (Show Source):
You can put this solution on YOUR website!
I need help factoring some equations please help:
2. 16xy + 32yz – 20xyz
3. 9x2 – 64
4. 5x3 – 125u2x
5. x2 + 7x + 12
Please explain to me how to do these because I am confused and old.


2. 16xy+32yz-20xyz. Take a look at all three terms and find what is common. That is, find the common factor.

2a. In this case, let's start with the common factor for the numbers 16, 32, -20. The common factor is 4. How did I calculate this? Find the lowest common factors: 16=2*2*2*2, 32=2*2*2*2*2, -20=-2*2*5. The common factor is 2*2=4. So factor out 4 in all three terms which yields: 4(4xy+8yz-5xyz)

2b. Now, check if x is common in all three terms. x is not a common factor. So it cannot be factored out.

2c. Now check if y is common to all three terms. y is a common factor. So factor out y. 4y(4x+8z-5xz).

2d. Now check if z is common to all three terms. z is not a common factor.

2e. Therefore, final answer is 4y(4x+8z-5xz).


3. 9x2 – 64.

3a. This is a special case of the FOIL method. That is, multiply (Ax-B)(Ax+B) where A=3 or Ax=3x and B=8. FOIL means F=First O=Outer, I=Inner, L=Last: Multiply First Terms (Ax*Ax), then Outer Terms, Ax*B, then Inner Terms, -B*Ax, then Last Terms, -B*B. Therefore,

3b. (Ax-B)(Ax+B)=Ax*Ax+Ax*B-B*Ax-B*B. We can simplify this since the INNER and OUTER terms add up to zero. Simplifying, yields (Ax-B)(Ax+B)=(Ax)^2-B^2.

3c. So 9x2-64=(3x-8)(3x+8) is the final answer where A=3 or Ax=3x and B=8.


4. 5x3 – 125u2x. Follows a similar process as Problem 2.

4a. 5x3-125u2x= 5(x3-25u2x). Factored common factor 5.
4b. 5(x3-25u2x)=5x(x2-25u2). Factors common factor x. 5x(x2-25u2) can be rewritten as 5x(x^2-25u^2), where u2 is meant as u-squared or u^2.


5. x2 + 7x + 12 x2+7x+12.

5a. Based on the FOIL method again, see above Problem 3. You need to satisfy two equations: A+B=7 and A*B=12. So you need to find factors of 12 that add to seven. You can use a trial and error method for this case.

5b. 2 and 6 are factors of twelve but they don't add to seven.
1 and 12 are factors of twelve but they don't add to seven.

5c. Now, let's try 3 and 4. When you multiply these numbers, you get 3*4=12. When you add these numbers, you get 3+4=7.

5d. Therefore, x2+7x+12=(x+3)(x+4) is the answer. Now, use FOIL method to check answer:

(x+3)(x+4)=x*x+4x+3x+3*4=x2+7x+12. So it works.


Note: If you prefer a visual approach to factoring, I have developed free step-by-step videos (1-2 hours) on Factoring at http://www.FreedomUniversity.TV/courses/IntroAlgebra/Module5.html. The set of videos have many similar problems as the ones that are describe above.





Distributive-associative-commutative-properties/211195: This question is from textbook
please help me solve this problem:
T=3vs-4ws+5vw solve for V please I got v= t+4ws/8sw
1 solutions
Answer 159594 by drj(1380) About Me  on 2009-09-06 10:24:14 (Show Source):
You can put this solution on YOUR website!
T=3vs-4ws+5vw solve for V please I got v= t+4ws/8sw
Step 1: Given: T=3vs-4ws+5vw
Step 2: Group terms containing v on right side of equation and other terms not containing v on left side. Here's how it works:
Step 2a. Add 4ws to both sides of equation: T+4ws=3vs-4ws+4ws+5vw. Note on right side of equation: -4ws+4ws=0 to simplify on next step
Step 2b. Simplify right side: T+4ws=3vs+5vw Again note we used: -4ws+4ws=0
Step 3. Now Solve for v by factoring and divide. Here's how it works:
Step 3a. Factor out v on right side: T+4ws=v(3s+5w)
Step 3b. You want to isolate v by itself. Now divide (3s+5w)in both sides of equations: (T+4ws)/(3s+5w)=v(3s+5w)/(3s+5w). Note: On right side (3s+4w)/(3s+5w)=1. So we can simplify in final step.
Step 3c. (T+4ws)/(3s+5w)=v
Step 4. We can rewrite final answer as v=(T+4ws)/(3s+5w)

Distributive-associative-commutative-properties/210620: This question is from textbook pre algebra A (a transition to algebra and geometry) (california edition)
Hi
I Im the 7th grade and im doing Pre Algebra homework.and i just cant seem to figure it out ,could u help me? the problem is (5+7)x and the directionS say to :restate each expression using the distributive Property.do not simplify.
and thats it. PLEASE HELP I AM IN DEEP DEEP TROUBLE!!!!

1 solutions
Answer 159587 by drj(1380) About Me  on 2009-09-06 10:04:03 (Show Source):
You can put this solution on YOUR website!
Step 1. Given (5+7)x
Step 2. Distributive property (5+7)x=5x+7x You can stop here.
Step 3. Simplify. 5x+7x=12x

Functions/209939: Which is greater: 1 and 1/4rth miles in 2 minutes 4 secs. OR 1 and 3/16th miles in 1 minute 55 secs.
Please show some work if possible. Ty.
_kat
1 solutions
Answer 158751 by drj(1380) About Me  on 2009-09-01 10:43:16 (Show Source):
You can put this solution on YOUR website!
Step 1. I assume you want to find which one has the greater velocity=distance/time.
Step 2.
a. Distance Calculation. 1 and 1/4 is equal to 1.25=1+(1/4) where 1/4=0.25. Distance=1.25 miles.

b. Time Calculation. Time is 2 minutes and 4 secs=2*60 seconds+4 secs=124 seconds where 60 seconds=1 minute.

c. Velocity Calculation. Velocity in miles per second=1.25 miles/124 seconds which is equal to about .010081 miles/sec=(0.010081 miles/sec)*(3600 sec/hour)=36.29 miles/hour where one hour=3600 seconds.


Step 3.

a. Distance Calculation. 1 and 3/16 is equal to 1+(3/16)=16/16+3/16=19/16 miles. I put the fraction in a common denominator and 16/16=1.


b. Time Calculation. Time is 1 minute and 55 seconds=60 seconds+55 seconds=115 seconds where 1 minute=60 seconds.

c. Velocity Calculation. Velocity in miles per second is (19/16 miles)/(115 seconds)= (0.010326 miles/second)*(3600 sec/hour)=37.17 miles/hour

Step 4. Step 3c has the greater velocity than Step 2c.