Questions on Algebra: Quadratic Equation answered by real tutors!

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Graph each number on the complex plane and find its absolute value..
8. 3 - 4i
Plot the point (3,-4); abs. value = sqrt(3^2 + 4^2) = 5
-----------------------
9. 8 + 10i
Plot the point (8,10); abs. value = sqrt(8^2+10^2) = 12.806...
-----------------------
10. 9 + 3i
Plot the point (9,3); abs. value = sqrt(9^2+3^2) = 9.4868...
====================
Cheers,
Stan H.

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The D they are referring to is the Determinant of the quad. It is found by using the coefficents of the standard form. When D<0 you get 2 complex roots for a solution.
:
D = b2 - 4ac
:
example of an equation that gives a negative D....
D = b2 - 4ac
= (-4)2 - 4(1)(13) = -36
:
solution

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . The discriminant -36 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 -36 is + or - . The solution is Here's your graph:

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if f(x)=ax^2+bx+c then
f has a solution if d=b^2-4ac >=0
s^2 - 4s + 4 = 0
d= (-4)^2-4(1)(4)=0 (has a solution with multiplicity 2)
5/6x2 - 7x - 6/5 = 0
d= (-7)^2-4(5/6)(-6/5)=53 (has 2 solutions)
7a2 + 8a + 2 = 0
d= 8^2-4(7)(2)=8 (has 2 solutions)

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Create a real-life situation that fits into the equation (x + 4)(x - 7) = 0 and express the situation as the same equation:
:
This would be one example:
:
The area of a rectangle is 28 sq ft,
where the width is 3 ft shorter than the Length.
:
width: W = (L - 3)
:
Area = L * (L-3) = 28
:
L^2 - 3L - 28 = 0
(L + 4)(L - 7) = 0
L = 7

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1. common factors 1st __ -8(x^2-4)

difference of squares __ -8(x+4)(x-4)


2. factors of 3 and -10 that "combine" to give 13 __ (3x-2)(x+5)


3. factors of 5 and -4 that "combine" to give -19 __ (5x+1)(x-4)

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1. factors of 4 and -15 that "combine" to give -17 __ (4x+3)(x-5)=0

2. factors of 10 and 6 that "combine" to give 19 __ (5x+2)(2x+3)=0



3. dividing by 3 __ x^2=1600

taking square root __ x=±40

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if r is a root (solution) of a quadratic equation, then x-r is a factor

(x-1)(x+8)=0 __ FOILing __ x^2+7x-8=0

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The first equation is a circle with radius 6 and the center is at (-2,4)
(x+2)^2 + (y-4)^2 = 36
The general equation is x^2+y^2= (radius)^2
This is a parabola
y=3x^2

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=0 is zero! That means that there is only one solution: . Expression can be factored: Again, the answer is: 0, 0. Here's your graph:

This is also a parabola
y=x^2+x

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=1 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 0, -1. Here's your graph:

This is also an equation for a circle
(x-1)^2 + (y-8)^2 = 16
The center is at (1,8) and the radius is 4.
I hope that this helps!

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Complete the square on

Step 1, starting from a quadratic in standard form like you have is to put the constant term on the right:



Step 2, if the coefficient on the term is other than 1, divide by that coefficient.



Step 3, divide the resulting coefficient on the term by 2 and square the result



Step 4, add this result to both sides of your equation and collect terms





Step 5, the above result has a perfect square on the left (hence the term "completing the square"), so factor it:



Step 6, take the square root of both sides:

or

Which leads us to a great big oops! because you can't take the square root of a negative number. The solution is to use the imaginary number which is defined as , leaving us with:

or
or
or

Step 7, isolate and simplify in each equation





If you want the exact representation of the roots of the given equation, you are done. If you need a numerical approximation of the imaginary parts of your complex numbers, get out your calculator.

Multiplying to verify that the product is, in fact, , is left as an exercise for the student. Alternatively, you could just trust me.

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You are correct. There are a number of ways to verify your answer, but you can plug in the answer back into the equation and simplify.

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Complete the square by adding 6-1/4:
---> Remember to subtract 6-1/4 too so the value of the expression does not change.
----> convert to improper fractions so it's easier to deal with.
, ANSWER
You know what, it follows "vertex-form" for a quadratic function:
--->
We have, , & --> VERTEX POINTS
Let's see the graph below:
---> see vertex (2-1/2,-81/4)
.
Let's see the x-intercepts:
--->
where----->
By Quadratic Eqn:




See again the graph:
---> see the x-intercepts (-2,0) & (7,0).It's good!
Thank you,
Jojo

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http://www.themathpage.com/alg/quadratic-equations.htm

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subtracting 8m __ m^2-8m=3

adding half of the coefficient of the 1st order term, squared __ m^2-8m+(-8/2)^2=3+(-8/2)^2

m^2-8m+16=3+16 __ taking square root __ m-4=±sqrt(19)

adding 4 __ m=4±sqrt(19)

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subtracting 5 __ x^2=8

taking square root __ x=±sqrt(8)

factoring out whole roots __ x=±2sqrt(2)

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3x^2 = 8
First divide both sides by 3
x^2=8/3
Take the square root of both numbers
x= sq root of 8 =2(sq rt 2)/sq rt of 3 note that 8=4*2 the square root of 4 is 2
Need to rationalize the denomiator multiply the numerator and denomiator by sq rt of 3
2sq rt of 6/3

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=48 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 1.86602540378444, 0.133974596215561. Here's your graph:

Replace the variable x with m

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solve this quadratic equation by completing the square.
a^2 - 4a + 5 = 0
---
a^2 - 4a + ? = -5 + ?
a^2 -4a + (-4/2)^2 = -5 + (-4/2)^2
a^2 - 4a + 4 = -5 + 4
a^2 - 4a + 4 = -1
(a-2)^2 = -1
a-2 = i or a-2 = -i
a = 2+i or a = 2-i
======================
Comment: If you have not studied complex numbers the
answer you would get would be "no solution".
======================
Cheers,
Stan H.

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solve this quadratic equation by completing the square.
x^2 + 4x - 12 = 0
----
x^2 + 4x + ? = 12 + ?
x^2 + 4x + (4/2)^2 = 12 + (4/2)^2
x^2 + 4x + 4 = 12 + 4
(x+2)^2 = 16
x+2 = 4 or x+2 = -4
x = 2 or x = -6
=====================
Cheers,
Stan H.

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well we know that distance equals rate times time. d=rt
:
In our case distance is the same under both scenarios 600 miles.
:
what is different is the speeds and thus the times as speed and time are inversely related.
:
so in the first trip lets call the rate r and the time t.
in the second scenario the rate would be r+20 and the time is t-1.
:
600=rt....................eq 1
600=(r+20)(t-1)...........eq 2
:
lets rewrite eq 1 as t=600/r and place that value into eq 2
:

multiply by r
multiply factors on right side of eq
combine like terms on one side of equation

:
throw out the negative speed
:
so speed on first trip is mph and on second trip is mph
:

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=48400 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 100, -120. Here's your graph:

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Start with the given equation.


Notice we have a quadratic equation in the form of where , , and


Since , which means that "a" is positive, this means that the parabola opens upwards and that the function has a minimum.

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I'm assuming that you want to solve


Start with the given equation.


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


or Break up the expression.


So the answers are or


which approximate to or

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=21 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 1.79128784747792, -2.79128784747792. Here's your graph:

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a=2
b=-3
c=1
:
since a is positive this opens upward and has a minimum

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Start with the given equation.


Notice we have a quadratic equation in the form of where , , and


Since (ie "a" is positive), this tells us that the parabola opens upward and the function has a minimum.

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You need to post the full problem. We don't have your book.

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Let s=speed of stream

Remember, the distance-rate-time formula is




Divide both sides by r to solve for t.


Rearrange the equation.


So when she goes upstream (against the current), the stream is slowing her down. So this means that and . So the equation for the upstream portion of the journey is:



When she goes downstream (with the current), the stream is speeding her up. So this means that and . So the equation for the downstream portion of the journey is:




Now simply add these two equations together to get the total time 3 hours like this:





Multiply both sides by the LCD to clear the fractions.


FOIL


Distribute


Combine like terms.


Subtract 243 from both sides.


Divide both sides by .


Take the square root of both sides. Note: only the positive square root is considered.

So the speed of the stream is 3 km/hr

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we know the area of a rectangle is Length times Width which we will label L and W. A=L*W
:
L=3W--->W=L/3
A=75
:
75=L(L/3)
:

:

:
length
:
width

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D=b2- 4ac if the discriminant is less than zero you will have complex roots.
:
where

example of negative determinant
:
D=(-10)^2-4(1)(34)=-36
solution 5+3i and 5-3i

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . The discriminant -36 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 -36 is + or - . The solution is Here's your graph:

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If , then the quadratic will have two complex (ie non real) solutions.


Why? Recall that the quadratic formula is




The discriminant D is everything in the square root. So this means that




Now if , this means that "D" is negative. Remember you CANNOT take the square root of a negative number. This is why you will NOT get any real solutions if


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

For example, let's find the discriminant for


From we can see that , , and


Start with the discriminant formula


Plug in , , and


Square to get


Multiply to get


Subtract from to get


Since the discriminant is less than zero, this means that there are two complex solutions



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



Now let's use the quadratic formula to find the solutions of


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Simplify to get . Note:


or Break up the expression.


or Break up the fraction for each case.


or Reduce.


So our answers are or


Since our answers are complex (ie not real), this verifies our original claim.

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a=1,b=-3,and c=-1

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Create a real-life situation that fits into the equation (x + 2)(x - 5) = 0 and express the situation as the same equation:
:
This would be one example:
:
The area of a rectangle is 10 sq ft,
where the width is 3 ft shorter than the Length.
:
width: W = (L - 3)
:
Area = L * (L-3) = 10
:
L^2 - 3L - 10 = 0
(L + 2)(L - 5) = 0
L = 5

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just reverse the process..if you have x=1 then what does it take to get x to equal zero.....subtract 1 from each side..x-1=0 do the same for x=-8...only this time you have to add 8...x+8=0

so we have or or

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a) x2 + 6x - 7 = 0
(x+7)(x-1)=0
x+7=0
x=-7 ans.
x-1=0
x=1 ans.
-------------------------
b) z2 + z + 1 = 0
z = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}
z=(-1+-sqrt[1^2-4*1*1])/2*1
z=(-1+-sqrt[1-4])/2
z=(-1+-sqrt-3)/2
z=(-1+-1.732)/2
z=-1/2+-.866i
----------------------------------------
I'll leave the rest for you to solve.
c) (3)1/2y2 - 4y - 7(3)1/2 = 0
d) 2x2 - 10x + 25 = 0
e) 2x2 - 6x + 5 = 0
f) s2 - 4s + 4 = 0
g) 5/6x2 - 7x - 6/5 = 0
h) 7a2 + 8a + 2 = 0

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Rita rows 12 km upstream and 12 km downstream in 3 hours. The speed of her boat in still water is 9 km/hr. Find the speed of the stream.
-------------------
Call the speed of the stream w (for water).
Going upstream, her speed (with respect to the land) is 9-w
Going downstream, it's 9+w
See that?
The time it takes is the distance (s) over the speed.
So upstream it's 12/(9-w) hours
Downstream is 12/(9+w) hours
Make sense so far?
Now, 12/(9-w) + 12/(9+w) = 3
It's no longer a "word problem", it's one equation in one unknown.
---------------
12/(9-w) + 12/(9+w) = 3
4/(9-w) + 4/(9+w) = 1
Multiply thru by (9-w)*(9+w)
4(9+w) + 4(9-w) = w^2 - 81
72 = w^2 - 81
w^2 = 9
w = 3 km/hr (also -3, but that makes no sense and can be discarded)

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Let s=speed of stream

Remember, the distance-rate-time formula is




Divide both sides by r to solve for t.


Rearrange the equation.


So when she goes upstream (against the current), the stream is slowing her down. So this means that and . So the equation for the upstream portion of the journey is:



When she goes downstream (with the current), the stream is speeding her up. So this means that and . So the equation for the downstream portion of the journey is:




Now simply add these two equations together to get the total time 3 hours like this:





Multiply both sides by the LCD to clear the fractions.


FOIL


Distribute


Combine like terms.


Subtract 243 from both sides.


Divide both sides by .


Take the square root of both sides. Note: only the positive square root is considered.

So the speed of the stream is 3 km/hr

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alright so we would normally find out an average cost per person by taking the cost and dividing by number of people....lets call the original # of people attending x so avg cost would be 500/x but now x is decreasing by 5 and the average cost is increasing by 5 so
:
mulitply all terms by x(x-5)
:

:
500x-2500=500x-5x^2+25x}}}
:
divide by 5
:

:

:
throw out the negative and we have the original amount of people of 25
:
therefore the number of people attending the party is
:

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=2025 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 25, -20. Here's your graph:

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Let c=cost per individual (the cost before the students failed to arrive) and n=number of students


So the total cost is divided among n people to get




Since 5 failed to attend, this means that students showed up. Now the new cost is




Start with the given equation.


Plug in


Multiply every term by the LCD . This cancels out every denominator.


Distribute


Subtract from both sides.


Subtract from both sides.


Let's use the quadratic formula to solve for n


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


Since a negative amount of people doesn't make any sense, this means that the only answer is


So 25 people were going to attend but only 20 actually attended.

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A designer, attempting to arrange the characters of his artwork in the form of a square grid with equal number of rows and columns, found that 24 characters were left out. When he tried to add one more row and column, he found that he was short of 25 characters. Can you find the number of characters used by the designer?
---------------------------------
Original arrangement DATA:
Let the number or rows and columns be "x"
# of characters is x^2 + 24
------------------------------
Changed arrangement DATA:
Number of rows and columns is "x+1"
# of characters is (x+1)^2 -25
-------------------------------------
EQUATION:
# of characters = # of characters
x^2 + 24 = (x+1)^2 - 25
x^2 + 24 = x^2 + 2x + 1 -25
24 = 2x - 24
2x = 48
x = 24
------------------
Ans: # of characters = 24^2+24 = 600
=========================================
Cheers,
Stan H.

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Write three quadratic equations, with a, b, and c (coefficients of x2, x, and the constant) as:
Integers : y = 2x^2 - 3x + 8
----------------------------------
Rational numbers : y = (1/2)x^2 - (5/8)x + 4
------------------------------------------------
Irrational numbers y = sqrt(3)x^2 + sqrt(7)x - cuberoot(10)
==============================================================
Cheers,
Stan H.

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Complete the square. It works like this:
Subtract from both sides

Now take one-half of the co-efficient of ,
square it and add this to both sides.


The left side is a perfect square. Rewrite it.


Now take the square root of both sides

answer 1
and, also, since square root is both (+) and (-)
answer 2

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Quadractic is polynomial of degree 2.A1*x^2+A2*x+A3 some like Ax^2 +Bx + C Notice (x-3)*(x-4) not= x^2 -7*X +12 'WHEN CAN YOU GET AWAY WITH SAYING IT IS????' (-3)*(-4) = (-1)^2*12 What's (-1)^2? When? What's (x^2-4)/(x-2) are u at level u "STAY AWAY FROM 'DIVISION BY 1-1"? wHY? cAN'T GENERALLY SOLVE FOR x 'complete square heh both work the "roots" are ,when have x^2+bx +c ,(-b+(or-)(b^2-4*c)^.5)/2 where b^2=>4*c OR NO ROOTS UNLESS SOME FANTASY!!DID YOU SEE SOMEWHERE 'DONE WITH FACTORING THEN PROCEEDS MORE"FACTORING" AND "FORMULAS" What did u think about that???Are u studying algebra for the 'grade' ? then stick it

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a)


Start with the given expression


Factor out the GCF


So factors to

---

b)


Start with the given expression


Factor out the GCF


So factors to

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I'll do the first two to get you started


# 1




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.


So FOILs to .


In other words, .

---

# 2




Start with the given expression.


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.


So FOILs to .


In other words, .