Quadratic Equations

What is a Quadratic Equation?

A quadratic equation in variable $x$ is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, $c$ are real numbers and $a \neq 0$. The word 'quadratic' comes from 'quadratum' (Latin for square) because the highest power of the variable is 2. Examples: $x^2 - 5x + 6 = 0$, $2x^2 + 3x = 0$, $x^2 = 25$. A root (or solution) is a value of $x$ that satisfies the equation.

Method 1: Solving by Factorisation

This is the fastest method when it works. The idea: write $ax^2 + bx + c$ as a product of two linear factors, then use the zero product property — if $AB = 0$, then $A = 0$ or $B = 0$.

Steps: (1) Write in standard form. (2) Find two numbers whose product = $ac$ and sum = $b$. (3) Split the middle term. (4) Factor by grouping. (5) Set each factor = 0.

Example: Solve $6x^2 - x - 2 = 0$. Here $ac = -12$, $b = -1$. Numbers: $3$ and $-4$ (product = $-12$, sum = $-1$). Split: $6x^2 + 3x - 4x - 2 = 0$. Factor: $3x(2x+1) - 2(2x+1) = 0$, so $(3x-2)(2x+1) = 0$. Roots: $x = 2/3$ or $x = -1/2$.

Method 2: Completing the Square

This method converts the equation into a perfect square form. Starting from $ax^2 + bx + c = 0$:

(1) Divide by $a$: $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$
(2) Move constant: $x^2 + \frac{b}{a}x = -\frac{c}{a}$
(3) Add $(\frac{b}{2a})^2$ to both sides: $\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$
(4) Take square root: $x + \frac{b}{2a} = \pm\frac{\sqrt{b^2-4ac}}{2a}$

This is actually how the quadratic formula is derived! You rarely need to complete the square in CBSE board exams, but understanding it helps you see WHERE the formula comes from.