\(\begin{aligned} x^2+6x&=16 \\ x&=2,\ x=-8 \end{aligned}\)

\(ax^2+bx+c\equiv px^2+qx+r\quad\Longrightarrow\quad a=p,\ b=q,\ c=r\)

Compare coefficients only when the equality holds for every x.

Example: \(3x^2-5x+4\equiv a(x-1)^2+b(x-1)+c\)

Choose values of \(x\) that make a bracket zero.

Let \(x=1\):
\(\begin{aligned} 3-5+4&=c \\ c&=2 \end{aligned}\)

Expand the right side, then match coefficients of each power of \(x\).

\(\begin{aligned} x^2:\ a&=3 \\ x:\ b-2a&=-5 \\ b&=1 \end{aligned}\)

Polynomial Identities and Comparing Coefficients