i) Find the Sum and Product of Roots

Use the formula:  Sum of roots = $\displaystyle -\frac{b}{a}$ and Product of roots = $\displaystyle \frac{c}{a}$, where a, b and c are the coefficients of the quadratic equation ax2+bx+c=0.

$\displaystyle \alpha +4\alpha =-\frac{p}{1}\quad \text{and}\quad \alpha \times \text{4}\alpha \text{=}\frac{q}{1}$

$\displaystyle \text{Sum}\ \text{of}\ \text{Roots}\ \text{= -p}\ \ \quad \text{Product}\ \text{of}\ \text{Roots = q}$

ii) Show that 4p2=25q

$\displaystyle 5\alpha =-p\to (1)\quad \quad \text{4}{{\alpha }^{2}}\text{=q}\to \text{(2)}$

Make $\displaystyle \alpha$ the subject of the formula (1).

$\displaystyle \alpha =\frac{{-p}}{5}\to (1)$

Substitute (1) into (2) and simplify the equation

$\displaystyle \text{4(}\frac{{-p}}{5}{{)}^{2}}\text{=q}$

$\displaystyle \text{4}\times \frac{{{{p}^{2}}}}{{25}}\text{=q}$

$\displaystyle \frac{{4{{p}^{2}}}}{{25}}\text{=q}$

$\displaystyle 4{{p}^{2}}\text{=25q}$

When you first start learning Sum and Products of  roots, the roots are almost always $\displaystyle \alpha$ and $\displaystyle \beta$. But as you progress toward  intermediate level, the roots could be like those in the example above or those below.

If one root is twice the other,
$\displaystyle \text{ }\!\!\alpha\!\!\text{ }\ \ \text{and}\ \text{2 }\!\!\alpha\!\!\text{ }\quad \text{or}\quad \beta \ \ \text{and}\ \text{2}\beta$

If two roots differs by two,
$\displaystyle \text{ }\!\!\alpha\!\!\text{ }\ \ \text{and}\ \text{ }\!\!\alpha\!\!\text{ +3}\quad \text{or}\quad \beta \ \ \text{and}\ \beta \text{+3}$

If one root is the reciprocal of the other,
$\displaystyle \text{ }\!\!\alpha\!\!\text{ }\ \ \text{and}\ \frac{1}{\alpha }\quad or\quad \beta \ \ \text{and}\ \frac{1}{\beta }$