WAEC Further Mathematics 2012 Paper

Concept:A quadratic inequality is solved by writing it in factored form and then checking the sign of the product in different intervals.Explanation:Start with the given inequality.$$x^{2} - 2x \ge 3$$Subtract $3$ from both sides.$$x^{2} - 2x - 3 \ge 0$$Factor the quadratic.$$(x - 3)(x + 1) \ge 0$$The boundary points are $x = 3$ and $x = -1$.For $x < -1$, for example $x = -2$: $(-2 - 3)(-2 + 1) = (-5)(-1) = 5 > 0$, so the inequality is true.For $-1 < x < 3$, for example $x = 0$: $(0 - 3)(0 + 1) = -3 < 0$, so the inequality is false.For $x > 3$, for example $x = 4$: $(4 - 3)(4 + 1) = 5 > 0$, so the inequality is true.At $x = -1$ and $x = 3$, the product is $0$, so both values satisfy the inequality.Therefore, the solution is all $x$ such that $x \le -1$ or $x \ge 3$.Answer:$$x \le -1 \quad \text{or} \quad x \ge 3$$