Problem Statement
If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $p(x) = 2x^2 - 5x + 7$, find the value of:
(i) $\alpha^2 + \beta^2$
(ii) $\frac{1}{\alpha} + \frac{1}{\beta}$
Verified Solution & Marking Scheme
Determine Sum and Product of Zeros
From Vieta's formulas for $p(x) = 2x^2 - 5x + 7$:
$\alpha + \beta = -\frac{b}{a} = -\frac{-5}{2} = \frac{5}{2}$
$\alpha\beta = \frac{c}{a} = \frac{7}{2}$
Part (i): Calculate α² + β²
$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = \left(\frac{5}{2}\right)^2 - 2\left(\frac{7}{2}\right) = \frac{25}{4} - 7 = \frac{25 - 28}{4} = -\frac{3}{4}$
Part (ii): Calculate 1/α + 1/β
$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{5/2}{7/2} = \frac{5}{7}$