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 Linear-systems/124659: Given f(x) = x2 + 5x + 2, find f(0).1 solutions Answer 91369 by jim_thompson5910(28504)   on 2008-02-03 19:02:19 (Show Source): You can put this solution on YOUR website! Start with the given function. Plug in . In other words, replace each x with 0. Evaluate to get 0. Multiply 5 and 0 to get 0 Now combine like terms
 Radicals/124660: Solve. x squared - 6x - 3 = 01 solutions Answer 91367 by jim_thompson5910(28504)   on 2008-02-03 19:01:10 (Show Source): You can put this solution on YOUR website!Let's use the quadratic formula to solve for x: Starting with the general quadratic the general solution using the quadratic equation is: So lets solve ( notice , , and ) Plug in a=1, b=-6, and c=-3 Negate -6 to get 6 Square -6 to get 36 (note: remember when you square -6, you must square the negative as well. This is because .) Multiply to get Combine like terms in the radicand (everything under the square root) Simplify the square root (note: If you need help with simplifying the square root, check out this solver) Multiply 2 and 1 to get 2 So now the expression breaks down into two parts or Now break up the fraction or Simplify or So these expressions approximate to or So our solutions are: or Notice when we graph , we get: when we use the root finder feature on a calculator, we find that and .So this verifies our answer
 Radicals/124661: Find the axis of symmetry of y = 2x squared + 8x - 71 solutions Answer 91366 by jim_thompson5910(28504)   on 2008-02-03 18:58:22 (Show Source): You can put this solution on YOUR website!To find the axis of symmetry, use this formula: From the equation we can see that a=2 and b=8 Plug in b=8 and a=2 Multiply 2 and 2 to get 4 Reduce So the axis of symmetry is
 Expressions-with-variables/124663: Please help me figure out the value of y. Problem: 3y + 2 < 7 - 2y1 solutions Answer 91364 by jim_thompson5910(28504)   on 2008-02-03 18:56:54 (Show Source): You can put this solution on YOUR website! Start with the given inequality Subtract 2 from both sides Add 2y to both sides Combine like terms on the left side Combine like terms on the right side Divide both sides by 5 to isolate y Divide -------------------------------------------------------------- Answer: So our answer is
 Graphs/124560: Hello, Please help! Solve by graphing: y= -3x+3 y= 2(x-2)-3 Thank you1 solutions Answer 91291 by jim_thompson5910(28504)   on 2008-02-03 13:23:00 (Show Source): You can put this solution on YOUR website! Start with the second equation Distribute Combine like terms So our new system is Now let's graph the first equation (note: if you need help with graphing, check out this solver) Graph of Now let's plot the second graph Graph of (red) and (green) From the graph, we can see that the two lines intersect at (2,-3). So the solution is and
 Graphs/124559: Hello, Please help solve 2x-5y=-3 y=-4x+1 Thank you 1 solutions Answer 91290 by jim_thompson5910(28504)   on 2008-02-03 13:14:12 (Show Source): You can put this solution on YOUR website! Start with the given system Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown. Distribute Combine like terms on the left side Add 5 to both sides Combine like terms on the right side Divide both sides by 22 to isolate x Reduce Now that we know that , we can plug this into to find Substitute for each Simplify So our answer is and which also looks like
Polynomials-and-rational-expressions/124554: 6x^2+5xy-21y^2
1 solutions

Answer 91284 by jim_thompson5910(28504)   on 2008-02-03 12:47:11 (Show Source):
You can put this solution on YOUR website!
Do you want to factor this?

Looking at we can see that the first term is and the last term is where the coefficients are 6 and -21 respectively.

Now multiply the first coefficient 6 and the last coefficient -21 to get -126. Now what two numbers multiply to -126 and add to the middle coefficient 5? Let's list all of the factors of -126:

Factors of -126:
1,2,3,6,7,9,14,18,21,42,63,126

-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-63,-126 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -126
(1)*(-126)
(2)*(-63)
(3)*(-42)
(6)*(-21)
(7)*(-18)
(9)*(-14)
(-1)*(126)
(-2)*(63)
(-3)*(42)
(-6)*(21)
(-7)*(18)
(-9)*(14)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5

First NumberSecond NumberSum
1-1261+(-126)=-125
2-632+(-63)=-61
3-423+(-42)=-39
6-216+(-21)=-15
7-187+(-18)=-11
9-149+(-14)=-5
-1126-1+126=125
-263-2+63=61
-342-3+42=39
-621-6+21=15
-718-7+18=11
-914-9+14=5

From this list we can see that -9 and 14 add up to 5 and multiply to -126

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

-------------------------------

So factors to

 Equations/124533: Hi Can I have some finding an equation of the line through (3, 4) with slope 5/6? Thanks1 solutions Answer 91274 by jim_thompson5910(28504)   on 2008-02-03 11:45:42 (Show Source): You can put this solution on YOUR website! If you want to find the equation of line with a given a slope of which goes through the point (,), you can simply use the point-slope formula to find the equation: ---Point-Slope Formula--- where is the slope, and is the given point So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Distribute Multiply and to get Add 4 to both sides to isolate y Combine like terms and to get (note: if you need help with combining fractions, check out this solver) ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line with a slope of which goes through the point (,) is: which is now in form where the slope is and the y-intercept is Notice if we graph the equation and plot the point (,), we get (note: if you need help with graphing, check out this solver) Graph of through the point (,) and we can see that the point lies on the line. Since we know the equation has a slope of and goes through the point (,), this verifies our answer.
 Linear-equations/124478: This question is from textbook Introductory Algebra Graph on a plane. y > -11 solutions Answer 91237 by jim_thompson5910(28504)   on 2008-02-02 23:42:47 (Show Source): You can put this solution on YOUR website!In order to graph , we need to graph the equation (just replace the inequality sign with an equal sign). So lets graph the line (simply draw a horizontal line through ) graph of Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality with the test point Substitute (0,0) into the inequality Plug in and Simplify (note: for some reason, some of the following images do not display correctly in Internet Explorer. So I recommend the use of Firefox to see these images.) Since this inequality is true, we simply shade the entire region that contains (0,0) Graph of with the boundary (which is the line in red) and the shaded region (in green) (note: since the inequality contains a greater-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line)
 Linear-equations/124483: Can someone help? I am understanding part of the problem but not the whole problem. I feel like I am missing something..thanks Solve each system by the substitution method. Determine whether the equations are independent, dependent, or inconsistent..Thanks first equation second equation1 solutions Answer 91236 by jim_thompson5910(28504)   on 2008-02-02 23:39:39 (Show Source): You can put this solution on YOUR website!Start with the given system Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown. Distribute Combine like terms on the left side Subtract 25 from both sides Combine like terms on the right side Divide both sides by -3 to isolate x Divide Now that we know that , we can plug this into to find Substitute for each Simplify So our answer is and which also looks like So because we got a unique solution, this means that the system is independent. Notice if we graph the two equations, we can see that their intersection is at . So this verifies our answer. Graph of (red) and (green)
 Equations/124417: Please help. (2-9x^3)^21 solutions Answer 91176 by jim_thompson5910(28504)   on 2008-02-02 16:47:12 (Show Source): You can put this solution on YOUR website! Start with the given expression Expand. Remember something like Now let's FOIL the expression Remember, when you FOIL an expression, you follow this procedure: Multiply the First terms: Multiply the Outer terms: Multiply the Inner terms: Multiply the Last terms: Now collect every term to make a single expression Now combine like terms Rearrange the terms in descending order --------------------- Answer: So foils and simplifies to In other words,
 Equations/124403: Hi can I have some help finding an equation of the line with x-intercept -1 and y-intercept 8?1 solutions Answer 91172 by jim_thompson5910(28504)   on 2008-02-02 16:12:47 (Show Source): You can put this solution on YOUR website!If the x-intercept is -1 and y-intercept is 8, then we have the point (-1,0) for the x-intercept and the point (0,8) for the y-intercept So let's find the equation of the line that goes through (-1,0) and (0,8) First lets find the slope through the points (,) and (,) Start with the slope formula (note: is the first point (,) and is the second point (,)) Plug in ,,, (these are the coordinates of given points) Subtract the terms in the numerator to get . Subtract the terms in the denominator to get Reduce So the slope is ------------------------------------------------ Now let's use the point-slope formula to find the equation of the line: ------Point-Slope Formula------ where is the slope, and is one of the given points So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Rewrite as Distribute Multiply and to get Add to both sides to isolate y Combine like terms and to get ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line which goes through the points (,) and (,) is: The equation is now in form (which is slope-intercept form) where the slope is and the y-intercept is Notice if we graph the equation and plot the points (,) and (,), we get this: (note: if you need help with graphing, check out this solver) Graph of through the points (,) and (,) Notice how the two points lie on the line. This graphically verifies our answer.
 Equations/124400: Please help. (x^2+x+6)(x-6) Thank you.1 solutions Answer 91171 by jim_thompson5910(28504)   on 2008-02-02 16:09:19 (Show Source): You can put this solution on YOUR website! Start with the given expression Rearrange the terms Expand the expression. Remember something like expands to Distribute Multiply Combine like terms Now rearrange the terms in descending order So expands and simplifies to . In other words, If you need more help with multiplying polynomials, check out this Long Multiplication Calculator
 Graphs/124412: Solve each of the following systems of linear inequalities graphically. 10. 2x + y ≤ 8 x + y ≥ 3 x ≥ 0 y ≥ 0 1 solutions Answer 91170 by jim_thompson5910(28504)   on 2008-02-02 15:58:09 (Show Source):
 Graphs/124410: Solve each of the following systems by substitution. 8. 3x – y = -15 x = y -7 1 solutions Answer 91168 by jim_thompson5910(28504)   on 2008-02-02 15:52:15 (Show Source): You can put this solution on YOUR website! Start with the given system Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown. Distribute Combine like terms on the left side Add 21 to both sides Combine like terms on the right side Divide both sides by 2 to isolate y Divide Now that we know that , we can plug this into to find Substitute for each Simplify So our answer is and which also looks like Notice if we graph the two equations, we can see that their intersection is at . So this verifies our answer. Graph of (red) and (green)
 Graphs/124408: Solve each of the following systems by graphing. 6. 4x + 3y = 12 x + y = 2 1 solutions Answer 91166 by jim_thompson5910(28504)   on 2008-02-02 15:47:33 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: In order to graph these equations, we need to solve for y for each equation. So let's solve for y on the first equation Start with the given equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce Now lets graph (note: if you need help with graphing, check out this solver) Graph of So let's solve for y on the second equation Start with the given equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce Now lets add the graph of to our first plot to get: Graph of (red) and (green) From the graph, we can see that the two lines intersect at the point (,) (note: you might have to adjust the window to see the intersection)
 Graphs/124407: Solve each of the following systems by graphing. 5. x – y = 8 x + y = 2 1 solutions Answer 91165 by jim_thompson5910(28504)   on 2008-02-02 15:45:48 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: In order to graph these equations, we need to solve for y for each equation. So let's solve for y on the first equation Start with the given equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce Now lets graph (note: if you need help with graphing, check out this solver) Graph of So let's solve for y on the second equation Start with the given equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce Now lets add the graph of to our first plot to get: Graph of (red) and (green) From the graph, we can see that the two lines intersect at the point (,) (note: you might have to adjust the window to see the intersection)
 Graphs/124405: Evaluate each function for the value specified 4. f (x)= x^3 + 2x^2 - 7x+ 9; find (a) f (-2), (b) f (0), and (c) f (2). 1 solutions Answer 91164 by jim_thompson5910(28504)   on 2008-02-02 15:42:50 (Show Source): You can put this solution on YOUR website! a) Let's evaluate Start with the given function. Plug in . In other words, replace each x with -2. Evaluate to get -8. Evaluate to get 4. Multiply 2 and 4 to get 8 Multiply -7 and -2 to get 14 Now combine like terms -----------Now let's evaluate another value--------- b) Let's evaluate Start with the given function. Plug in . In other words, replace each x with 0. Evaluate to get 0. Evaluate to get 0. Multiply 2 and 0 to get 0 Multiply -7 and 0 to get 0 Now combine like terms -----------Now let's evaluate another value--------- c) Let's evaluate Start with the given function. Plug in . In other words, replace each x with 2. Evaluate to get 8. Evaluate to get 4. Multiply 2 and 4 to get 8 Multiply -7 and 2 to get -14 Now combine like terms
 Graphs/124404: Evaluate each function for the value specified 3. f (x) = -2x^3 + 5x^2 - x - 1; find (a) f (-1), (b) f (0), and (c) f (2). 1 solutions Answer 91163 by jim_thompson5910(28504)   on 2008-02-02 15:40:41 (Show Source): You can put this solution on YOUR website!a) Let's evaluate Start with the given function. Plug in . In other words, replace each x with -1. Evaluate to get -1. Evaluate to get 1. Multiply -2 and -1 to get 2 Multiply 5 and 1 to get 5 Multiply -1 and -1 to get 1 Now combine like terms -----------Now let's evaluate another value--------- b) Let's evaluate Start with the given function. Plug in . In other words, replace each x with 0. Evaluate to get 0. Evaluate to get 0. Multiply -2 and 0 to get 0 Multiply 5 and 0 to get 0 Multiply -1 and 0 to get 0 Now combine like terms -----------Now let's evaluate another value--------- c) Let's evaluate Start with the given function. Plug in . In other words, replace each x with 2. Evaluate to get 8. Evaluate to get 4. Multiply -2 and 8 to get -16 Multiply 5 and 4 to get 20 Multiply -1 and 2 to get -2 Now combine like terms
 Graphs/124402: Graph each of the following inequalities 2. 4x + 3y > 12 1 solutions Answer 91162 by jim_thompson5910(28504)   on 2008-02-02 15:38:12 (Show Source): You can put this solution on YOUR website!In order to graph , we need to graph the equation (just replace the inequality sign with an equal sign). So lets graph the line (note: if you need help with graphing, check out this solver) graph of Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality with the test point Substitute (0,0) into the inequality Plug in and Simplify (note: for some reason, some of the following images do not display correctly in Internet Explorer. So I recommend the use of Firefox to see these images.) Since this inequality is not true, we do not shade the entire region that contains (0,0). So this means we shade the region that is on the opposite side of the line Graph of with the boundary (which is the line in red) and the shaded region (in green) (note: since the inequality contains a greater-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line)
 Graphs/124401: Graph each of the following inequalities 1. x – y ≥ 4 1 solutions Answer 91161 by jim_thompson5910(28504)   on 2008-02-02 15:36:01 (Show Source): You can put this solution on YOUR website!In order to graph , we need to graph the equation (just replace the inequality sign with an equal sign). So lets graph the line (note: if you need help with graphing, check out this solver) graph of Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality with the test point Substitute (0,0) into the inequality Plug in and Simplify Since this inequality is not true, we do not shade the entire region that contains (0,0). So this means we shade the region that is on the opposite side of the line Graph of with the boundary (which is the line in red) and the shaded region (in green)
 Linear-equations/124397: Hi! Can I have some help on finding an equation of the line through (1, 5) with slope 2? 1 solutions Answer 91160 by jim_thompson5910(28504)   on 2008-02-02 14:52:51 (Show Source): You can put this solution on YOUR website! If you want to find the equation of line with a given a slope of which goes through the point (,), you can simply use the point-slope formula to find the equation: ---Point-Slope Formula--- where is the slope, and is the given point So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Distribute Multiply and to get Add 5 to both sides to isolate y Combine like terms and to get ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line with a slope of which goes through the point (,) is: which is now in form where the slope is and the y-intercept is Notice if we graph the equation and plot the point (,), we get (note: if you need help with graphing, check out this solver) Graph of through the point (,) and we can see that the point lies on the line. Since we know the equation has a slope of and goes through the point (,), this verifies our answer.
 Equations/124398: This question is from textbook College Algebra Sullivan I need to see the solution worked out in steps. Thank you X^3-9X+=01 solutions Answer 91159 by jim_thompson5910(28504)   on 2008-02-02 14:52:01 (Show Source): You can put this solution on YOUR website! Start with the given expression Factor out the GCF Now set each factor equal to zero: or or Now solve for x in : So our solutions are: ,, or
The last line of synthetic division for (2x^4-5x^3+7x^2-3x+1)/(x-3) gives coefficients for the quotient of 2,1,8,21.
Is this true or false?
1 solutions

Answer 91148 by jim_thompson5910(28504)   on 2008-02-02 12:02:03 (Show Source):
You can put this solution on YOUR website!
Let's simplify this expression using synthetic division

First lets find our test zero:

Set the denominator equal to zero

Solve for x.

so our test zero is 3

Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.
 3 | 2 -5 7 -3 1 |

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 2)
 3 | 2 -5 7 -3 1 | 2

Multiply 3 by 2 and place the product (which is 6) right underneath the second coefficient (which is -5)
 3 | 2 -5 7 -3 1 | 6 2

Add 6 and -5 to get 1. Place the sum right underneath 6.
 3 | 2 -5 7 -3 1 | 6 2 1

Multiply 3 by 1 and place the product (which is 3) right underneath the third coefficient (which is 7)
 3 | 2 -5 7 -3 1 | 6 3 2 1

Add 3 and 7 to get 10. Place the sum right underneath 3.
 3 | 2 -5 7 -3 1 | 6 3 2 1 10

Multiply 3 by 10 and place the product (which is 30) right underneath the fourth coefficient (which is -3)
 3 | 2 -5 7 -3 1 | 6 3 30 2 1 10

Add 30 and -3 to get 27. Place the sum right underneath 30.
 3 | 2 -5 7 -3 1 | 6 3 30 2 1 10 27

Multiply 3 by 27 and place the product (which is 81) right underneath the fifth coefficient (which is 1)
 3 | 2 -5 7 -3 1 | 6 3 30 81 2 1 10 27

Add 81 and 1 to get 82. Place the sum right underneath 81.
 3 | 2 -5 7 -3 1 | 6 3 30 81 2 1 10 27 82

Since the last column adds to 82, we have a remainder of 82. This means is not a factor of
Now lets look at the bottom row of coefficients:

Looking at the last line, we see the coefficients: 2,1,10, and 27

So the coefficients for the quotient are not 2,1,8, and 21

So the answer is false and you are correct.

 Graphs/124382: Graph the solution to the inequality │y – 1│< 21 solutions Answer 91147 by jim_thompson5910(28504)   on 2008-02-02 11:56:32 (Show Source): You can put this solution on YOUR website! Start with the given inequality Break up the absolute value (remember, if you have , then and ) and Break up the absolute value inequality using the given rule Combine the two inequalities to get a compound inequality Add 1 to all sides ---------------------------------------------------- Answer: So our answer is which looks like this in interval notation if you wanted to graph the solution set, you would get Graph of the solution set in blue and the excluded values represented by open circles
 Square-cubic-other-roots/124320: Thanks for your help ^4 sqrt 811 solutions Answer 91115 by jim_thompson5910(28504)   on 2008-02-01 21:14:28 (Show Source): You can put this solution on YOUR website! Start with the given expression Rewrite the radical expression using exponent notation Factor into Rewrite the expression using the identity Convert the expression back to radical notation Take the square root of 81 to get 9 Take the square root of 9 to get 3 So