Question

How do you factor a perfect square trinomial `9x^2 − 12x + 4`?

Answer 1

`9x^2-12x+4=(3x-2)^2`

Explanation:

If `Ax^2+Bx+C = (ax+c)^2` is a perfect square trinomial
then for some set `{a,c}`
`color(white)("XXXX")` `A=a^2`
`color(white)("XXXX")` `C=c^2`
and
`color(white)("XXXX")` `B = 2ac`

Given that `9x^2-12x+4` is a perfect square trinomial
`color(white)("XXXX")` `A=9x^2 rarr a=3x`
`color(white)("XXXX")` `C=4 rarr c=+-2`

Since `B=-12x = 2ac = 2(3x)c`
`color(white)("XXXX")` `rarr c=-2`