# Additional Math – Quadratic Equations – Find the Equation with roots alpha & 4 alpha

i) Find the Sum and Product of Roots

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

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

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

ii) Show that 4p2=25q

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

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

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

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

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

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

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

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

When you first start learning Sum and Products of  roots, the roots are almost always $latex \displaystyle \alpha$ and $latex \displaystyle \beta$. 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,
$latex \displaystyle \text{ }\!\!\alpha\!\!\text{ }\ \ \text{and}\ \text{2 }\!\!\alpha\!\!\text{ }\quad \text{or}\quad \beta \ \ \text{and}\ \text{2}\beta$

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

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

## Share:

### Math – Statistics – 1st Quartile, Median and 3rd Quartile Ungroup Data

The above two diagrams show you how to find the 1st quartile, median (2nd Quartile) and the 3rd quartile ungroup

### Additional Math – Differentiation – Quotient Rule (Challenging)

Differentiate $latex \displaystyle\ y=\frac{{{{x}^{2}}\sqrt{{x+1}}}}{{x-1}}$ with respect to x. Simplify the Numerator (otherwise you need to use both quotient rule for

### Additional Math – Binomial theorem – Using Normal Expansion vs Binomial Theorem

The above video shows  two method of expanding the  expression; using the algebraic expansion (rainbow method) versus the binomial theorem.