Quadratic Equations and Inequalities

2.1

Quadratic Equations and Inequalities

Definition of Quadratic Equation

A quadratic equation in general form can be wirtten as
$ax^2+bx+c=0$
where

$a$, $b$ and $c$ are constants and $a \neq0$.

$The image is an educational graphic titled ‘Methods of Solving Quadratic Equations.’ It features two methods: 1. ‘Completing the Square Method’: - Rewrite $ ax^2 + bx + c = 0 $ in the form of $ a(x - h)^2 + k = 0 $. - Solve for $ x $. 2. ‘Formula Method’: - Use the formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. The graphic includes the Pandai logo at the bottom right corner.$

Roots of Quadratic Equation

Solutions or roots of the quadratic equation $ax^2+bx+c=0$ are the $x$-coordinates of the intersection points between the graph
$y=ax^2+bx+c$ and the $x$-axis.

Formula for Solving Quadratic Equation

The formula for solving a quadratic equation $ax^2+bx+c=0$ is given as:

$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

Forming Quadratic Equation from Given Roots

Formula for Sum and Product of Roots

Formula for sum of roots is given by:

$\text{Sum of roots}=\alpha+\beta=-\dfrac{b}{a}$

While formula of product of roots is given by:

$\text{Product of roots}=\alpha\beta=\dfrac{c}{a}$

Forming Quadratic Equation

The quadratic equation with roots $\alpha$ and $\beta$ can be written as:

$x^2-(\text{sum of roots})x+(\text{product of roots})=0$

Example $1$

Question

Solve the equation $x^2+4x-7=0$ by using completing the square method.

Solution

Based on equation $x^2+4x-7=0$,

$a=1$,
$b=4$,
$c=-7$.

Move the constant term, $c$ to the right hand side of the equation,

$\begin{aligned} x^2+4x-7&=0 \\ x^2+4x&=7. \end{aligned}$

Add the term $\left( \dfrac{b}{2} \right)^2$ on the left and right hand side of the equation,

$\begin{aligned} x^2+4x+\left( \dfrac{4}{2} \right)^2&=7+\left( \dfrac{4}{2} \right)^2 \\ x^2+4x+2^2&=7+2^2\\ (x+2)^2&=11\\ x+2&=\pm \sqrt{11}. \end{aligned}$

$x=-5.317$ or $x=1.317$.

Hence, the solutions of the equation $x^2+4x-7=0$ are $-5.317$ and $1.317$.

Example $2$

Question

Solve the equation $2x^2-2x-3=0$ by using formula.

Solution

Based on the equation $2x^2-2x-3=0$,

$a=2$,
$b=-2$,
$c=-3$.

Use the formula for solving quadratic equation,

$\begin{aligned} x&=\dfrac{-(-2)\pm \sqrt{(-2)^2-4(2)(-3)}}{2(2)} \\ &=\dfrac{2\pm \sqrt{28}}{4} \end{aligned}$

$\begin{aligned} x&=\dfrac{2-\sqrt{28}}{4} \\ &=-0.823 \end{aligned}$ or $\begin{aligned} x&=\dfrac{2+\sqrt{28}}{4} \\ &=1.823 .\end{aligned}$

Hence, the solutions of the equation $2x^2-2x-3=0$ are $-0.823$ and $1.823$.