What is the rule for simplying $ax^2 + bx + c$ for $a \neq 1$

Question

I have a trivial quadratic

$-3x^2 + 4x-4/3$

What is the most direct way to show it is equivalent to

$-(3x - 2)^2/3$

Confusion:

I used the quadratic formula and found the root to be $2/3$, and usually, you would end it at $(x - 2/3)^2$. But here our leading coefficient is not 1, therefore you cannot do this.

What is the general way for putting $ax^2 + bx + c$ into $\gamma(\alpha x + \beta)^2$??

Answer 1

You can, however, write any parabola $ax^2 + bx + c$ in the form $$a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a},$$ called the vertex form of the parabola, since $$\left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)$$ is the vertex of the parabola, i.e. the maximum (if $a < 0$) or minimum (if $a > 0$) point of the parabola