SOLUTION: Solve by completing the square. 2x^2 – 6x – 3 = 0

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Question 103658: Solve by completing the square. 2x^2 – 6x – 3 = 0
Found 2 solutions by ankor@dixie-net.com, jim_thompson5910:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
Solve by completing the square.
:
2x^2 – 6x – 3 = 0
:
When completing the square, we need the coefficient of x^2 to be; divide eq by 2:
x^2 - 3x - 3%2F2 = 0
:
x^2 - 3x + ___ = 3%2F2
:
Find the value that will make it a perfect square, divide the coefficient of x by 2 and square it; add to both sides:
%283%2F2%29%5E2 = 9%2F4
we have:
x^2 - 3x + 9%2F4 = 3%2F2 + 9%2F4
:
x^2 - 3x + 9%2F4 = 6%2F4 + 9%2F4; use the common denominator on the right
:
x^2 - 3x + 9%2F4 = +15%2F4
:
(x - 3%2F2)^2 = 15%2F4
:
x - 3%2F2 = +/- sqrt%2815%2F4%29; square root of both sides
:
x - 3%2F2 = +/- %281%2F2%29sqrt%2815%29; extract the square root of 1/4
:
x = 3%2F2 +/- %281%2F2%29sqrt%2815%29; Add 3%2F2 to both sides
Same as:
x = %283+%2B-+sqrt%2815%29%29%2F2;
Two solutions:
x = %283+%2B+sqrt%2815%29%29%2F2
and
x = %283+-+sqrt%2815%29%29%2F2

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

2x%5E2-6x-3=0 Start with the given equation


2x%5E2-6x=3 Add 3 to both sides
2%28x%5E2-3x%29=3 Factor out the leading coefficient 2. This step is important since we want the x%5E2 coefficient to be equal to 1.



Take half of the x coefficient -3 to get -1.5 (ie -3%2F2=-1.5)
Now square -1.5 to get 2.25 (ie %28-1.5%29%5E2=2.25)



2%28x%5E2-3x%2B2.25%29=3%2B2.25%282%29 Add this result (2.25) to the expression x%5E2-3x inside the parenthesis. Now the expression x%5E2-3x%2B2.25 is a perfect square trinomial. Now add the result (2.25)(2) (remember we factored out a 2) to the right side.



2%28x-1.5%29%5E2=3%2B2.25%282%29 Factor x%5E2-3x%2B2.25 into %28x-1.5%29%5E2


2%28x-1.5%29%5E2=3%2B4.5 Multiply 2.25 and 2 to get 4.5



2%28x-1.5%29%5E2=7.5 Combine like terms on the right side

%28x-1.5%29%5E2=3.75 Divide both sides by 2


x-1.5=0%2B-sqrt%283.75%29 Take the square root of both sides

x=1.5%2B-sqrt%283.75%29 Add 1.5 to both sides to isolate x.

So the expression breaks down to
x=1.5%2Bsqrt%283.75%29 or x=1.5-sqrt%283.75%29


So our answer is approximately
x=3.43649167310371 or x=-0.436491673103709

Here is visual proof

+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+2x%5E2-6x-3%29+ graph of y=2x%5E2-6x-3


When we use the root finder feature on a calculator, we would find that the x-intercepts are x=3.43649167310371 and x=-0.436491673103709, so this verifies our answer.