Question

P(x)=21+24x+6x^2 in vertex form

Answer 1

We want vertex form:  y-k=(x-h)^2.

Re-write   P(x)=21+24x+6x^2    as    P(x)=6x^2 + 24x + 21

Factor the first 2 terms as shown:     P(x)=6(x^2+4x           )+ 21

complete the square of (x^2+4x):     P(x) =6(x^2 + 4x + 4 - 4) + 21

Rewrite the quadratic:                         P(x) = 6(x+2)^2 - 24 + 21

Simplify this result:                                P(x) = 6(x+2)^2 - 3
Compare this to                                     y-k = a(x-h)^2

Identify k as -3 and h as -2; vertex is at (-2,-3)

P(x) in vertex form is        P(x) = 6(x+2)^2 - 3