SOLUTION: Solve by completing the square. Show your work.[please] 29. 2x2 - 6x + 1 = 0 30. -x2 - 8x + 5 = 0 31. 9x2 - 18x - 1 = 0 32. -4x2 + 8x - 3 = 0

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Solve by completing the square. Show your work.[please] 29. 2x2 - 6x + 1 = 0 30. -x2 - 8x + 5 = 0 31. 9x2 - 18x - 1 = 0 32. -4x2 + 8x - 3 = 0      Log On


   

Question 177853: Solve by completing the square. Show your work.[please]

29. 2x2 - 6x + 1 = 0
30. -x2 - 8x + 5 = 0

31. 9x2 - 18x - 1 = 0
32. -4x2 + 8x - 3 = 0
Found 2 solutions by jim_thompson5910, gonzo:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first two to get you started

29)



2x%5E2-6x%2B1 Start with the given expression.


2%28x%5E2-3x%2B1%2F2%29 Factor out the x%5E2 coefficient 2. This step is very important: the x%5E2 coefficient must be equal to 1.


Take half of the x coefficient -3 to get -3%2F2. In other words, %281%2F2%29%28-3%29=-3%2F2.


Now square -3%2F2 to get 9%2F4. In other words, %28-3%2F2%29%5E2=%28-3%2F2%29%28-3%2F2%29=9%2F4


2%28x%5E2-3x%2Bhighlight%289%2F4-9%2F4%29%2B1%2F2%29 Now add and subtract 9%2F4 inside the parenthesis. Make sure to place this after the "x" term. Notice how 9%2F4-9%2F4=0. So the expression is not changed.


2%28%28x%5E2-3x%2B9%2F4%29-9%2F4%2B1%2F2%29 Group the first three terms.


2%28%28x-3%2F2%29%5E2-9%2F4%2B1%2F2%29 Factor x%5E2-3x%2B9%2F4 to get %28x-3%2F2%29%5E2.


2%28%28x-3%2F2%29%5E2-7%2F4%29 Combine like terms.


2%28x-3%2F2%29%5E2%2B2%28-7%2F4%29 Distribute.


2%28x-3%2F2%29%5E2-7%2F2 Multiply.


So after completing the square, 2x%5E2-6x%2B1 transforms to 2%28x-3%2F2%29%5E2-7%2F2. So 2x%5E2-6x%2B1=2%28x-3%2F2%29%5E2-7%2F2.


So 2x%5E2-6x%2B1=0 is equivalent to 2%28x-3%2F2%29%5E2-7%2F2=0.


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

Now let's solve 2%28x-3%2F2%29%5E2-7%2F2=0


2%28x-3%2F2%29%5E2-7%2F2=0 Start with the given equation.


2%28x-3%2F2%29%5E2=0%2B7%2F2Add 7%2F2 to both sides.


2%28x-3%2F2%29%5E2=7%2F2 Combine like terms.


%28x-3%2F2%29%5E2=%287%2F2%29%2F%282%29 Divide both sides by 2.


%28x-3%2F2%29%5E2=7%2F4 Reduce.


x-3%2F2=0%2B-sqrt%287%2F4%29 Take the square root of both sides.


x-3%2F2=sqrt%287%2F4%29 or x-3%2F2=-sqrt%287%2F4%29 Break up the "plus/minus" to form two equations.


x-3%2F2=sqrt%287%29%2F2 or x-3%2F2=-sqrt%287%29%2F2 Simplify the square root.


x=3%2F2%2Bsqrt%287%29%2F2 or x=3%2F2-sqrt%287%29%2F2 Add 3%2F2 to both sides.


x=%283%2Bsqrt%287%29%29%2F%282%29 or x=%283-sqrt%287%29%29%2F%282%29 Combine the fractions.


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Answer:


So the solutions are x=%283%2Bsqrt%287%29%29%2F%282%29 or x=%283-sqrt%287%29%29%2F%282%29.







30)





-x%5E2-8x%2B5 Start with the given expression.


-%28x%5E2%2B8x-5%29 Factor out the x%5E2 coefficient -1. This step is very important: the x%5E2 coefficient must be equal to 1.


Take half of the x coefficient 8 to get 4. In other words, %281%2F2%29%288%29=4.


Now square 4 to get 16. In other words, %284%29%5E2=%284%29%284%29=16


-%28x%5E2%2B8x%2Bhighlight%2816-16%29-5%29 Now add and subtract 16 inside the parenthesis. Make sure to place this after the "x" term. Notice how 16-16=0. So the expression is not changed.


-%28%28x%5E2%2B8x%2B16%29-16-5%29 Group the first three terms.


-%28%28x%2B4%29%5E2-16-5%29 Factor x%5E2%2B8x%2B16 to get %28x%2B4%29%5E2.


-%28%28x%2B4%29%5E2-21%29 Combine like terms.


-%28x%2B4%29%5E2%2B21 Distribute.



So after completing the square, -x%5E2-8x%2B5 transforms to -%28x%2B4%29%5E2%2B21. So -x%5E2-8x%2B5=-%28x%2B4%29%5E2%2B21.


So -x%5E2-8x%2B5=0 is equivalent to -%28x%2B4%29%5E2%2B21=0.



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Now let's solve -%28x%2B4%29%5E2%2B21=0


-%28x%2B4%29%5E2%2B21=0 Start with the given equation.


-%28x%2B4%29%5E2=0-21Subtract 21 from both sides.


-%28x%2B4%29%5E2=-21 Combine like terms.


%28x%2B4%29%5E2=%28-21%29%2F%28-1%29 Divide both sides by -1.


%28x%2B4%29%5E2=21 Reduce.


x%2B4=0%2B-sqrt%2821%29 Take the square root of both sides.


x%2B4=sqrt%2821%29 or x%2B4=-sqrt%2821%29 Break up the "plus/minus" to form two equations.


x=-4%2Bsqrt%2821%29 or x=-4-sqrt%2821%29 Subtract 4 from both sides.


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Answer:


So the solutions are x=-4%2Bsqrt%2821%29 or x=-4-sqrt%2821%29.

Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
equation # 29 is: ******************************************************
2%2Ax%5E2+-+6%2Ax+%2B+1+=+0
standard form of the equation is:
ax^2 + bx + c = 0
this is already in standard form.
a term = 2
b term = (-6)
c term = 1
---
first you move the constant to the right side of the equation by subtracting 1 from both sides of the equation to get :
2%2Ax%5E2+-+6%2Ax+=+-1
---
you then divide both sides of the equation by 2 to get:
x%5E2+-+3%2Ax+=+%28-1%2F2%29
---
you then take half of 3 and factor the left side of the equation to get:
%28x+-+%283%2F2%29%29%5E2+-+%283%2F2%29%5E2+=+%28-%281%2F2%29%29
this takes a little explaining.
start of explanation.
here's an example (not anything to do with this problem because the numbers are changed to make it simple).
take x%5E2+%2B+2x.
if you take half the 2 and make this equal to %28x%2B1%29%5E2, the answer will be:
x%5E2+%2B+2x+%2B+%281%29%5E2
that %281%29%5E2 is extra, so you have to subtract it to keep the original equality intact.
you get:
%28x%2B1%29%5E2+-+1+=+%28x%5E2+%2B+2x+%2B+1%29-+1+=+x%5E2+%2B+2x which is what you started off with.
this is exactly what we did above:
we took x%5E2+-+3%2Ax and factored it to get:
%28x-3%2F2%29%5E2+-+%283%2F2%29%5E2
if you do the multiplication, you will see that:

end of explanation.
---
you then add the (3/2)^2 term to both sides of the equation to get:
%28x+-+%283%2F2%29%29%5E2+=+%28-%281%2F2%29%29+%2B+%283%2F2%29%5E2
---
you then take the square root of both sides of the equation to get:
x+-+%283%2F2%29 = +/- sqrt%28%28-1%2F2%29+%2B+%283%2F2%29%5E2%29
---
you then add ((3/2)) to both sides of the equation to get:
x = +/- %28sqrt%28%28-1%2F2%29+%2B+%283%2F2%29%5E2%29%29+%2B+%283%2F2%29
---
after doing the math (i used a calculator), you will get:
x = 2.8228...
or
x = .1771...
---
to prove these values are correct, substitute them in the original equation (again i used the calculator with the full rather than the truncated values)
i took the original equation of 2x^2 - 6x + 1 = 0
and substituted these values to get:
0 = 0 both times proving both values are good.
---
equation number 32 is: ***********************************************
-4%2Ax%5E2+%2B+8%2Ax+-+3+=+0
standard form of the equation is:
ax^2 + bx + c = 0
this is already in standard form.
a term = -4
b term = 8
c term = -3
---
first you move the constant to the right side of the equation by adding 3 to both sides of the equation to get :
-4%2Ax%5E2+%2B+8%2Ax+=+3
---
you then divide both sides of the equation by (-4) to get:
x%5E2+-+2%2Ax+=+-3%2F4
---
you then take half of 2 and factor the left side of the equation to get:
%28x+-+1%29%5E2+-+%281%29%5E2+=+-3%2F4%29
---
you then add the (1)^2 term to both sides of the equation to get:
%28x-1%29%5E2+=+%28-3%2F4%29+%2B+1
---
you then take the square root of both sides of the equation to get:
x-1 = +/- sqrt%28%28-3%2F4%29+%2B+1%29
---
you then add 1 to both sides of the equation to get:
x = +/- %28sqrt%28%28-3%2F4%29+%2B+1%29%29+%2B+1
---
after doing the math (i used a calculator), you will get:
x = 1.5
or
x = .5
---
to prove these values are correct, substitute them in the original equation (again i used the calculator with the full rather than the truncated values)
i took the original equation of -4x^2 + 8x - 3 = 0
and substituted these values to get:
0 = 0 both times proving both values are good.
---
i will do number 30 next and i will leave number 31 for you to do.
if you follow the steps and understand what is going on, you should be able to complete it.
---
equation number 30 is: **************************************************
-x%5E2+-+8%2Ax+%2B+5+=+0
standard form of the equation is:
ax^2 + bx + c = 0
this is already in standard form.
a term is -1.
b term is -8.
c term is 5
---
first you move the constant to the right side of the equation by subtracting 5 from both sides of the equation to get :
-x%5E2+-+8%2Ax+=+-5
---
you then divide both sides of the equation by -1 to get:
x%5E2+%2B+8%2Ax+=+5
---
you then take half of 8 and factor the left side of the equation to get:
%28x+%2B+4%29%5E2+-+4%5E2+=+5
---
you then add the 4^2 term to both sides of the equation to get:
%28x%2B4%29%5E2+=+5+%2B+4%5E2
---
you then take the square root of both sides of the equation to get:
x%2B4+=+sqrt%285+%2B+4%5E2%29
---
you then subtract 4 from both sides of the equation to get:
x = +/- sqrt%285+%2B+4%5E2%29-4
---
after doing the math (i used a calculator), you will get:
x = .5828...
or
x = =-8.5825...
---
to prove these values are correct, substitute them in the original equation (again i used the calculator with the full rather than the truncated values)
i took the original equation of -x^2 - 8x + 5 = 0
and substituted these values to get:
0 = 0 both times proving both values are good.
---
by now, you should be able to do number 31 by yourself.
let me know if you are having problems.