Question

Factorize each of the following quadratic polynomials by using the method of completing the square:
z² − 4z − 12

Answer 1

\[z^2 - 4z - 12\]
\[ = z^2 - 4z + \left( \frac{4}{2} \right)^2 - \left( \frac{4}{2} \right)^2 - 12 [\text{ Adding and subtracting }\left( \frac{4}{2} \right)^2 ,\text{ that is }, 2^2 ]\]
\[ = z^2 - 4z + 2^2 - 2^2 - 12\]
\[ = (z - 2 )^2 - 16 [\text{ Completing the square }]\]
\[ = (z - 2 )^2 - 4^2 \]
\[ = [(z - 2) - 4][(z - 2) + 4]\]
\[ = (z - 6)(z + 2)\]