Question

How do you write `f(x)=2x^2+12x+12` in vertex form?

Answer 1

`y=2(x+3)^2-6`

Vertex form is also known as completing the square

Explanation:

`color(blue)("Step 1")`

Write as:

`y=2(x^2+6x)+12+k`

When we start to change things the equation becomes untrue. So we need to introduce the correction `k` to compensate for this. The value of `k` is calculated at the end.

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`color(blue)("Step 2")`

Move the power of 2 from `x^2` to outside the brackets.

`y=2(x+6x)^2+12+k`

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`color(blue)("Step 3")`

Halve the 6 from `6x`

`y=2(x+3x)^2+12+k`

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`color(blue)("Step 3")`

Remove the `x` from `3x`

`y=2(x+3)^2+12+k` .....................Equation(1)
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`color(blue)("Step 4")`

Determine the value of `k`

If you were to square the bracket we would have `color(magenta)(3^2)` form `color(green)(2)(x+ color(magenta)(3))^2`. Also this is multiplied by the `color(green)(2)` from outside the bracket giving:

`color(green)(2xx)color(magenta)(3^2) larr" this is the error we introduced"`

So we write:

`color(green)(2xx)color(magenta)(3^2)+k=0`

`=> 18+k=0 -> k=-18`

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`color(blue)("Step 5 - The final equation")`

Substitute this into equation(1)

`y=2(x+3)^2+12-18`

`y=2(x+3)^2-6`

`color(red)("I have superimposed both graphs so that you can see they are the same.")`