Example

If the roots of $x^2-3x-7=0$   are α  and β  , find the equation whose roots are ${\alpha }^2\beta$   and $\alpha {\beta }^2$ .

Solution

Part 1 : Find SoR and PoR for the quadratic equation in the question




Part 2 : Form a new quadratic equation by finding SoR and PoR

Example
The roots of $2x^2+3x-1=0$   are α and β.  Find the values of
(a) $(\alpha +1)(\beta +1)$
(b) $\frac{1}{\alpha }+\frac{1}{\beta }$
(c) ${\alpha }^2\beta +\alpha {\beta }^2$
(d) $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$

[Clue: ${\alpha }^2+{\beta }^2={(\alpha +\beta )}^2-2\alpha \beta$   ]

2.6 Finding the Sum of Roots (SoR) and Product of Roots (PoR)

Example
Find the sums and products of the roots of the following equations.
a. $x^2+7x-<!--\; -\; -->3=0$
b. $x(x-<!--\; -\; -->1)=5(1-<!--\; -\; -->x)$

Answer:
(a)
$\begin{array}{l}{x}^2+7x-3=0\\ a=1,b=7,c=-3\\ \\ \text{Sum of Roots}\\ \alpha +\beta =-\frac{b}{a}=-\frac{7}{1}=-7\\ \text{Product of Roots}\\ \alpha \beta =\frac{c}{a}=\frac{-3}{1}=-3\end{array}$

(b)
$\begin{array}{l}x(x-1)=5(1-x)\\ x^2-x=5-5x\\ x^2-x+5x-5=0\\ x^2+4x-5=0\\ a=1,b=4,c=-5\\ \\ \text{Sum of Roots}\\ \alpha +\beta =-\frac{b}{a}=-\frac{4}{1}=-4\\ \\ \text{Product of Roots}\\ \alpha \beta =\frac{c}{a}=\frac{-5}{1}-5\end{array}$

The quadratic equation $a{x}^2+bx+c=0$ can be solved by using the quadratic formula

$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Example 
Use the quadratic formula to find the solutions of the following equations.
a. $x^2+5x-<!--\; -\; -->24=0$
b. $x\left(x+4\right)=10$

Answer
(a)
For the equation $x^2+5x-<!--\; -\; -->24=0$
$\begin{array}{l}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-(5)\pm \sqrt{{(5)}^2-4(1)(-24)}}{2(1)}\\ x=\frac{-5\pm \sqrt{121}}{2}\\ x=-8\text{ or }x=3\end{array}$

(b)
$\begin{array}{l}x\left(x+\text{ 4}\right)=\text{ 10}\\ {\text{x}}^2+4x-10=0\\ \\ a=1,b=4,c=-10\\ \\ x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-(4)\pm \sqrt{{(4)}^2-4(1)(-10)}}{2(1)}\\ x=\frac{-4\pm \sqrt{56}}{2}\\ x=1.742\text{ or }x=-5.742\end{array}$