Quadratic Equations

Equations involving a squared variable, solved by factoring or the quadratic formula.

y = 1.0x² + 0.0x − 2.0

a (opens up/down) = 1.0b (shifts left/right) = 0.0c (y-intercept) = -2.0

Definition

A quadratic equation has the form $ax^2 + bx + c = 0$ where $a \neq 0$ . The highest power of $x$ is $2$ .

The two most common solution methods:

Factoring: rewrite as $(x - r)(x - s) = 0$ , then $x = r$ or $x = s$ .

Quadratic formula: always works:

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

Key properties

Solving by factoring

Solve $x^2 - 5x + 6 = 0$ .

Find two numbers that multiply to $6$ and add to $-5$ : that's $-2$ and $-3$ .

$(x - 2)(x - 3) = 0 \implies x = 2 \text{ or } x = 3$

Solving with the quadratic formula

Solve $2x^2 + 3x - 2 = 0$ .

$x = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm 5}{4}$

$x = \dfrac{1}{2}$ or $x = -2$

Common mistakes

Forgetting the $\pm$ : the formula always gives two candidates — don't drop one
Sign errors on $b$ : the formula starts with $-b$ , not $b$
Dividing only part of the numerator: $\frac{-b \pm \sqrt{\Delta}}{2a}$ means the whole numerator over $2a$

Try it

Solve $x^2 - 4 = 0$ .

Solution

$x^2 = 4 \implies x = \pm 2$

Or factor: $(x-2)(x+2) = 0$ .

Related concepts

Needs first