WAEC 2017 · Paper 2 · Q1

(a)
Find the range of values of $m$ for which $mx^2 + 2mx - 3 = 0$ has real roots.
Show the answer
$m \le -3$ or $m > 0$

Worked solution (try it first)

Read off the coefficients: $a = m$ , $b = 2m$ , $c = -3$ .
Real roots need $b^2 - 4ac \ge 0$ : $(2m)^2 - 4(m)(-3) \ge 0$ , so $4m^2 + 12m \ge 0$ .
Factorise: $4m(m + 3) \ge 0$ , with roots $m = 0$ and $m = -3$ .
" $\ge 0$ " is outside the roots: $m \le -3$ or $m \ge 0$ .
But $m = 0$ removes the $x^2$ term and leaves $-3 = 0$ , which has no roots at all.
So $m \le -3$ or $m > 0$ .