Find the constants $a,b,c$ to satisfy the re-writing of a function in another form.

Question

Given $f(x) = 3x^2 - 4x + 5$

what are the constants a, b, c if $f(x)$ was written in the form:

$a(x-b)^2 + c$

Answer 1

$$3x^2-4x+5 = a(x-b)^2+c \\$$

$$3x^2-4x+5 = ax^2-2abx+b^2+c \\$$

So, it has to be:

$$a=3,\\-2ab=-4,\\b^2+c=5$$

or:

$$a=3, b=\frac{2}{3}, c=\frac{41}{9}.$$