Question

Factorize each of the following quadratic polynomials by using the method of completing the square:
q2 − 10q + 21

Answer 1

\[q^2 - 10q + 21\]
\[ = q^2 - 10q + \left( \frac{10}{2} \right)^2 - \left( \frac{10}{2} \right)^2 + 21 [\text{ Adding and subtracting }\left( \frac{10}{2} \right)^2 ,\text{ that is }, 5^2 ]\]
\[ = q^2 - 2 \times q \times 5 + 5^2 - 5^2 + 21\]
\[ = (q - 5 )^2 - 4 [\text{ Completing the square }]\]
\[ = (q - 5 )^2 - 2^2 \]
\[ = [(q - 5) - 2][(q - 5) + 2]\]
\[ = (q - 5 - 2)(q - 5 + 2)\]
\[ = (q - 7)(q - 3)\]