What is the first step when rewriting y=-4x^2+2x-7 in the form y=a(x-h)^2+k?

3 Answers
Jul 3, 2017

There is a process for completing the square but the values, a,h, and k are far too easy to obtain by other methods. Please see the explanation.

Explanation:

  1. a = -4 the value of "a" is always the leading coefficient of the x^2 term.
  2. h=-b/(2a) = -2/(2(-4)) = 1/4
  3. k = y(h) = y(1/4) = -4(1/4)^2+2(1/4)-7 = -27/4

This is a lot easier than adding zero to the original equation in the form of -4h^2+4h^2:

y = -4x^2+2x-4h^2+4h^2-7

Removing a factor of -4 from the first 3 terms:

y = -4(x^2-1/2x+h^2)+4h^2-7

Match the middle term of the expansion (x-h)^2=x^2-2hx+h^2 with the middle term in the parenthesis:

-2hx = -1/2x

Solve for h:

h = 1/4

Therefore, we can compress the 3 terms into (x-1/4)^2:

y = -4(x-1/4)^2+4h^2-7

Substitute for h:

y = -4(x-1/4)^2+4(1/4)^2-7

Combine like terms:

y = -4(x-1/4)^2-27/4

Look at how much easier is it to remember 3 simple facts.

Jul 3, 2017

You would factor out the -4 from the first term giving you
y=-4(x^2-1/2x)-7

Explanation:

First complete the square.
y=-4x^2+2x-7
get the x^2 term to have a coefficient of 1.
You can do this by factoring out -4 from the first two terms.
y=-4(x^2-1/2x)-7
Then complete the square
y=-4(x-1/4)^2-7-(1/16xx-4)

this simplifies down to
y=-4(x-1/4)^2-6.75

Jul 3, 2017

Factor out -4 from each term, to get:

y = -4[x^2-1/2x+7/4]

Explanation:

y = ax^2 +bx+c

In order to complete the square, the coefficient of x^2 must be 1, so the first step will be to make this happen.

y = -4x^2 +2x-7" "larr factor out -4 from each term to get:

y = -4[x^2-1/2x+7/4]

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For the sake of completeness the full process is shown below.
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color(blue)(y = -4[x^2-1/2x" "+7/4])" "larr add and subtract (b/2)^2

b= -1/2" "rArr color(red)((b/2)^2 = (-1/2 div 2)^2 =(-1/4)^2 = 1/16)

color(blue)(y = -4[x^2-1/2x color(red)(+1/16 - 1/16)color(blue)(+7/4)])

y = -4[(x^2-1/2x +1/16)+( - 1/16+7/4)]

y = -4[(x-1/4)^2 +27/16]" "larr distribute the -4

y = -4(x-1/4)^2 -27/4

y = -4(x-1/4)^2 - 6 3/4