Question

Factorize each of the following quadratic polynomials by using the method of completing the square:
a² + 2a − 3

Answer 1

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