Completely Factor Quadratic Expression $ x^2 - 4x - 21 $

Question

Factorise the expression completely: $ x^2 - 4x - 21 $

Accepted Answer

$ (x + 3)(x - 7) $

Answer

Step 1: Recall that to factor $ x^2 + bx + c $ (or in this case $ x^2 - 4x - 21 $), we need two numbers $ m $ and $ n $ such that $ m \times n = -21 $ and $ m + n = -4 $. Step 2: List the factor pairs of $ -21 $: $ (1, -21) $, $ (-1, 21) $, $ (3, -7) $, $ (-3, 7) $. Step 3: Check which pair adds up to $ -4 $. The pair $ 3 $ and $ -7 $ satisfies $ 3 + (-7) = -4 $ and $ 3 \times (-7) = -21 $. Step 4: Rewrite the middle term using these numbers and factor by grouping (or directly write the factors): $ x^2 - 4x - 21 = (x + 3)(x - 7) $.