Question

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

Answer 1

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.

Answer 2

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`

Answer 3

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`