WAEC 2019 · Paper 2 · Q3

(a)
A binary operation $*$ is defined on the set of real numbers, $\mathbb R$ , by $x * y = x + y - 3xy$ , where $x, y \in \mathbb R$ . Find the identity element in $\mathbb R$ under the operation $*$ .
(b)
Find the range of values of $n$ for which $3 + 14n - 5n^2 \le 0$ .
Show the answer
$n \le -\frac15$ or $n \ge 3$

Worked solution (try it first)

Make the $n^2$ term positive: multiply by $-1$ and turn the sign round: $5n^2 - 14n - 3 \ge 0$ .
Factorise: two numbers that multiply to $5 \times (-3) = -15$ and add to $-14$ are $-15$ and 1.
So $5n^2 - 15n + n - 3 = 5n(n - 3) + 1(n - 3)$
$= (5n + 1)(n - 3) \ge 0$ .
The roots are $n = -\frac15$ and $n = 3$ . " $\ge 0$ " is outside them: $n \le -\frac15$ or $n \ge 3$ .
Check with $n = 0$ , between the roots: $3 + 0 - 0 = 3$ , which is not $\le 0$ ✓.