WAEC Further Mathematics 2023 Paper

Concept:For a quadratic equation, the sum and product of the roots can be obtained directly from its coefficients. These values can then be substituted into the required expression.Explanation:The given equation is $7x^2 + 12x - 4 = 0$.Comparing with the general form $ax^2 + bx + c = 0$, we get $a = 7$, $b = 12$, and $c = -4$.For roots $\alpha$ and $\beta$:Sum of roots: $\alpha + \beta = -\frac{b}{a} = -\frac{12}{7}$.Product of roots: $\alpha\beta = \frac{c}{a} = -\frac{4}{7}$.Now substitute these into $\frac{\alpha\beta}{(\alpha + \beta)^2}$:$$\frac{-\frac{4}{7}}{\left(-\frac{12}{7}\right)^2} = \frac{-\frac{4}{7}}{\frac{144}{49}}.$$Simplify by multiplying by the reciprocal:$$-\frac{4}{7} \times \frac{49}{144} = -\frac{4 \times 7}{144} = -\frac{28}{144}.$$Reduce the fraction:$$-\frac{28}{144} = -\frac{7}{36}.$$Therefore, the value is $-\frac{7}{36}$.Answer:$-\frac{7}{36}$, which corresponds to Option D.