Question

Write 3x^2-18x-6 in vertex form

Answer 1

The standard form of a quadratic equation is
`\displaystyle{ y=ax^2+bx+c`, while the vertex form is:


`y=a(x-h)^2+k`, where (h, k) is the vertex of the parabola.

What we want is to write
`\displaystyle{ y=3x^2-18x-6` as
`y=a(x-h)^2+k`

First, we note that all the three terms have a factor of 3, so we factorize it and write:


`\displaystyle{ y=3(x^2-6x-2)`.


Second, we notice that
`x^2-6x` are the terms produced by
`(x-3)^2=x^2-6x+9`, without the 9. So we can write:


`x^2-6x=(x-3)^2-9`, and substituting in
`\displaystyle{ y=3(x^2-6x-2)` we have:


`\displaystyle{ y=3(x^2-6x-2)=3[(x-3)^2-9-2]=3[(x-3)^2-11]`.

Finally, distributing 3 over the two terms in the brackets we have:


`y=3[x-3]^2-33`.


Answer:
`y=3(x-3)^2-33`