Quadratic Equations

What a quadratic equation is

A quadratic equation can be written as $ax^2 + bx + c = 0$ with $a \neq 0$. It has at most two roots.

Three ways to solve it

1. Factorisation. Best when the numbers are friendly.

$$x^2 - 5x + 6 = 0 \Rightarrow (x - 2)(x - 3) = 0 \Rightarrow x = 2 \text{ or } x = 3$$

2. Quadratic formula. Always works.

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

3. Graphical method. The roots are the $x$-intercepts of $y = ax^2 + bx + c$.

Common mistake: in $x(x - 3) = 2(x - 3)$, dividing both sides by $(x - 3)$ loses the root $x = 3$. Move everything to one side and factorise instead: $(x - 3)(x - 2) = 0$.

The discriminant

$\Delta = b^2 - 4ac$ tells you the nature of the roots without solving:

$\Delta$	Roots	Graph of $y = ax^2 + bx + c$
$\Delta > 0$	two distinct real roots	cuts the $x$-axis twice
$\Delta = 0$	one double real root (equal roots)	touches the $x$-axis
$\Delta < 0$	no real roots	does not meet the $x$-axis

When $\Delta < 0$, the roots are complex numbers of the form $a + bi$, where $i^2 = -1$.

Sum and product of roots

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$:

$$\alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}$$

A quadratic equation with roots $\alpha$ and $\beta$ is

$$x^2 - (\alpha + \beta)x + \alpha\beta = 0$$

Useful identities in exam questions:

• $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$
• $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$
• $\dfrac{1}{\alpha} + \dfrac{1}{\beta} = \dfrac{\alpha + \beta}{\alpha\beta}$

Exam tip: "Equal roots" means $\Delta = 0$; "real roots" means $\Delta \geq 0$; "two distinct real roots" means $\Delta > 0$. Read the wording carefully.