Answer:
The vertex of this parabola, [tex](-2, 3)[/tex], can be found by completing the square.
Step-by-step explanation:
The goal is to express this parabola in its vertex form:
[tex]g(x) = a\, (x - h)^2 + k[/tex],
where [tex]a[/tex], [tex]h[/tex], and [tex]k[/tex] are constants. Once these three constants were found, it can be concluded that the vertex of this parabola is at [tex](h,\, k)[/tex].
The vertex form can be expanded to obtain:
[tex]\begin{aligned}g(x)&= a\, (x - h)^2 + k \\ &= a\, \left(x^2 - 2\, x\, h + h^2\right) + k = a\, x^2 - 2\, a\, h\, x + \left(a\,h^2 + k\right)\end{aligned}[/tex].
Compare that expression with the given equation of this parabola. The constant term, the coefficient for [tex]x[/tex], and the coefficient for [tex]x^2[/tex] should all match accordingly. That is:
[tex]\left\lbrace\begin{aligned}& a = 3 \\ & -2\,a\, h = 12 \\& a\, h^2 + k = 15\end{aligned}\right.[/tex].
The first equation implies that [tex]a[/tex] is equal to [tex]3[/tex]. Hence, replace the "[tex]a\![/tex]" in the second equation with [tex]3\![/tex] to eliminate [tex]\! a[/tex]:
[tex](-2\times 3)\, h = 12[/tex].
[tex]h = -2[/tex].
Similarly, replace the "[tex]a[/tex]" and the "[tex]h[/tex]" in the third equation with [tex]3[/tex] and [tex](-2)[/tex], respectively:
[tex]3 \times (-2)^2 + k = 15[/tex].
[tex]k = 3[/tex].
Therefore, [tex]g(x) = 3\, x^2 + 12\, x + 15[/tex] would be equivalent to [tex]g(x) = 3\, (x - (-2))^2 + 3[/tex]. The vertex of this parabola would thus be:
[tex]\begin{aligned}&(-2, \, 3)\\ &\phantom{(}\uparrow \phantom{,\,} \uparrow \phantom{)} \\ &\phantom{(}\; h \phantom{,\,} \;\;k\end{aligned}[/tex].
