Question

Factorize each of the following quadratic polynomials by using the method of  completing the square:
p² + 6p + 8

Answer 1

\[p^2 + 6p + 8\]
\[ = p^2 + 6p + \left( \frac{6}{2} \right)^2 - \left( \frac{6}{2} \right)^2 + 8 [\text{ Adding and subtracting }\left( \frac{6}{2} \right)^2 , \text{ that is }, 3^2 ]\]
\[ = p^2 + 6p + 3^2 - 3^2 + 8\]
\[ = p^2 + 2 \times p \times 3 + 3^2 - 9 + 8\]
\[ = p^2 + 2 \times p \times 3 + 3^2 - 1\]
\[ = (p + 3 )^2 - 1^2 [\text{ Completing the square }]\]
\[ = [(p + 3) - 1][(p + 3) + 1]\]
\[ = (p + 3 - 1)(p + 3 + 1)\]
\[ = (p + 2)(p + 4)\]