Question

Factorize the following quadratic polynomial by using the method of completing the square:

$q^2-10q+21$

Answer 1

Given: $q^2-10q+21$

Since, the coefficient of $q^2$ is unity.

$\therefore$ Adding and subtracting $(\frac{1}{2}\times$ coefficient of $q$)${}^2$

i.e., $(–5{)}^2=(5{)}^2$

$\Rightarrow q^2–10q+21$

$=q^2–10q+5^2–5^2+21$

$=(q^2–10q+5^2)-25+21$

$=(q^2–10q+5^2)-4$

$=(q–5{)}^2–4$

$=(q–5{)}^2–2^2$

$=(q–5–2)(q–5+2)$

$=(q–7)(q–3)$

Hence, $q^2-10q+21=(q–7)(q–3)$