Question:
Factorize each of the following quadratic polynomials by using the method of completing the square:
y2 − 7y + 12
Solution:
$y^{2}-7 y+12$
$=y^{2}-7 y+\left(\frac{7}{2}\right)^{2}-\left(\frac{7}{2}\right)^{2}+12 \quad\left[\right.$ Adding and subtracting $\left.\left(\frac{7}{2}\right)^{2}\right]$
$=\left(y-\frac{7}{2}\right)^{2}-\frac{49}{4}+\frac{48}{4} \quad[$ Completing the square $]$
$=\left(y-\frac{7}{2}\right)^{2}-\frac{1}{4}$
$=\left(y-\frac{7}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}$
$=\left[\left(y-\frac{7}{2}\right)-\frac{1}{2}\right]\left[\left(y-\frac{7}{2}\right)+\frac{1}{2}\right]$
$=\left(y-\frac{7}{2}-\frac{1}{2}\right)\left(y-\frac{7}{2}+\frac{1}{2}\right)$
$=(y-4)(y-3)$