WAEC 2019 · Paper 2 · Q4

(a)
If $\alpha$ and $\beta$ are the roots of the equation $3x^2 + 4x - 5 = 0$ , find the value of $(\alpha - \beta)$ , leaving the answer in surd form.
Show the answer
$\pm\dfrac{2\sqrt{19}}{3}$

Worked solution (try it first)

For $3x^2 + 4x - 5 = 0$ : $\alpha + \beta = -\frac43$ and $\alpha\beta = -\frac53$ .
Square the difference: $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$ .
Substitute: $\frac{16}{9} - 4\left(-\frac53\right) = \frac{16}{9} + \frac{60}{9}$
$= \frac{76}{9}$ .
Take the square root: $\alpha - \beta = \pm\dfrac{\sqrt{76}}{3}$ .
Simplify the surd: $\sqrt{76} = \sqrt{4 \times 19}$
$= 2\sqrt{19}$ , so $\alpha - \beta = \pm\dfrac{2\sqrt{19}}{3}$ .