Question

Factorize each of the following algebraic expression:
(a + 7)(a − 10) + 16

Answer 1

\[(a + 7)(a - 10) + 16\]
\[ = a^2 - 10a + 7a - 70 + 16\]
\[ = a^2 - 3a - 54\]
\[\text{ To factorise }a^2 - 3a - 54 ,\text{ we will find two numbers p and q such that }p + q = - 3\text{ and }pq = - 54 . \]
Now, 
\[6 + ( - 9) = - 3 \]
and 
\[6 \times ( - 9) = - 54\]
\[\text{ Splitting the middle term }- 3a \text{ in the given quadratic as } - 9a + 6a, \text{ we get: }\]
\[ a^2 - 3a - 54 = a^2 - 9a + 6a - 54\]
\[ = ( a^2 - 9a) + (6a - 54)\]
\[ = a(a - 9) + 6(a - 9)\]
\[ = (a + 6)(a - 9)\]