SOLUTION: Write the following equation in vertex form using completing the square: y=2x^2 - x - 1 This is what I've tried so far, but I don't think I'm doing it right. y = (2x^2 - x +1/4)

Algebra ->  College  -> Linear Algebra -> SOLUTION: Write the following equation in vertex form using completing the square: y=2x^2 - x - 1 This is what I've tried so far, but I don't think I'm doing it right. y = (2x^2 - x +1/4)      Log On


   

Question 809394: Write the following equation in vertex form using completing the square:
y=2x^2 - x - 1
This is what I've tried so far, but I don't think I'm doing it right.
y = (2x^2 - x +1/4) - 1 - 1/4
Found 2 solutions by KMST, Edwin McCravy:
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
y=2x%5E2+-+x+-+1
y%2B1=2x%5E2+-+x
y%2B1=2%28x%5E2+-+x%2F2%29
x%5E2-x%2F2 is part of %28x-1%2F4%29%5E2=x%5E2-x%2F2%2B1%2F16 ,
so we add 2%2A%281%2F16%29 to both sides of the equal sign in y%2B1=2%28x%5E2+-+x%2F2%29 and get
y%2B1%2B2%2F16=2%28x%5E2+-+x%2F2%29%2A2%2A%281%2F16%29
y%2B1%2B1%2F8=2%28x%5E2+-+x%2F2%2B1%2F16%29
y%2B9%2F8=2%28x+-+1%2F4%29%5E2
highlight%28y=2%28x+-+1%2F4%29%5E2-9%2F8%29

Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
There are two ways, by completing the square:

First way:

    y = 2x² - x - 1

Get the constant term on the left by adding 1 to both sides:

y + 1 = 2x² - x

To get the first term on the right to just x² by multiplying
every term through by 1%2F2

1%2F2y + 1%2F2 = x² - 1%2F2x

Complete the square on the right side of the equation:
1.  Multiply the coefficient of x which is -1%2F2 by 1%2F2, getting -1%2F4
2.  Square that amount %281%2F4%29%5E2 = 1%2F16

3. Add that amount 1%2F16 to both sides of the equation:

1%2F2y + 1%2F2 + 1%2F16= x² - 1%2F2x + 1%2F16


Factor the right side as a perfect square:

1%2F2y + 1%2F2 + 1%2F16= (x - 1%2F4)²

Clear of fractions by multiplying every term on both sides by 16

8y + 8 + 1 = 16(x - 1%2F4)²

Combine 8 + 1 as 9

8y + 9 = 16(x - 1%2F4)

Subtract 9 from both sides

8y = 16(x - 1%2F4)² - 9

Solve for y by dividing every term by 8

 y = 2(x - 1%2F4)² - 9%2F8

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Second way:

    y = 2x² - x - 1

Factor 2 out of the first two terms:

    y = 2(x² - 1%2F2x) - 1

Change the parentheses to brackets so they can hold parentheses:

    y = 2[x² - 1%2F2x] - 1

Complete the square in the parentheses:
1.  Multiply the coefficient of x which is -1%2F2 by 1%2F2, getting -1%2F4
2.  Square that amount %281%2F4%29%5E2 = 1%2F16

3. Add and subtract that amount 1%2F16 inside the brackets:

    y = 2[x² - 1%2F2x + 1%2F16 - 1%2F16] - 1

Factor the first three terms in the bracket as a perfect square:

    y = 2[(x - 1%2F4)² - 1%2F16] - 1

Remove the bracket by distributing the 2, leaving the parentheses intact:

    y = 2(x - 1%2F4)² - 2%2Aexpr%281%2F16%29 - 1

    y = 2(x - 1%2F4)² - 1%2F8 - 1

Combine the last two terms, by writing 1 as 8%2F8

    y = 2(x - 1%2F4)² - 1%2F8 - 8%2F8

    y = 2(x - 1%2F4)² - 9%2F8

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Take your pick.

Edwin