Mathematics 9709/13 — October/November 2013

Walkthrough

Start by writing the quadratic as a product of two linear factors:

This factorisation is useful because the sign of the product changes only at the roots. The roots are $x = -1$ and $x = 2$. Since the coefficient of $x^2$ is positive, the parabola $y = x^2 - x - 2$ opens upwards. It is above the $x$-axis (positive) when $x$ is to the left of the smaller root or to the right of the larger root. Hence the solution is $x < -1$ or $x > 2$.

Key Takeaways

This question tests the ability to solve a quadratic inequality by factorising, finding roots, and using the sign of the quadratic. Understanding the shape of a positive-leading-coefficient parabola helps decide the intervals quickly.

Common Mistakes

• Writing the answer with $\leq$ or $\geq$ instead of strict inequalities, because the original inequality is strict ($>$).
• Reversing the intervals and writing $-1 < x < 2$, which would be the solution of $x^2 - x - 2 < 0$.
• Forgetting to factorise and trying to solve the inequality directly without considering the critical values.

Things to Be Careful About

The roots must be excluded because the inequality is strict. If the quadratic had a negative leading coefficient, the intervals would be reversed. Always confirm the sign by testing a value in each interval if unsure.