Question

Factorize each of the following quadratic polynomials by using the method of completing the square:
a² − 14a − 51

Answer 1

\[a^2 - 14a - 51\]
\[ = a^2 - 14a + \left( \frac{14}{2} \right)^2 - \left( \frac{14}{2} \right)^2 - 51 [\text{ Adding and subtracting }\left( \frac{14}{2} \right)^2 ,\text{ that is }, 7^2 ]\]
\[ = a^2 - 14a + 7^2 - 7^2 - 51\]
\[ = (a - 7 )^2 - 100 [\text{ Completing the square }]\]
\[ = (a - 7 )^2 - {10}^2 \]
\[ = [(a - 7) - 10][(a - 7) + 10]\]
\[ = (a - 7 - 10)(a - 7 + 10)\]
\[ = (a - 17)(a + 3)\]