How to Solve Quadratic Equations — Complete Guide

What Is a Quadratic Equation?

A quadratic equation is any equation that can be written in the form:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0.

Quadratic equations appear everywhere in maths and science — from projectile motion in physics to optimisation problems in real life.

Three Methods to Solve Quadratics

Method 1: Factoring

Factoring is the fastest method when it works. You're looking for two numbers that multiply to give ac and add to give b.

Example: Solve x² + 5x + 6 = 0

We need two numbers that multiply to 6 and add to 5
Those numbers are 2 and 3
So: (x + 2)(x + 3) = 0
Therefore: x = -2 or x = -3

Method 2: The Quadratic Formula

When factoring doesn't work easily, use the formula:

x = (-b ± √(b² - 4ac)) / 2a

This works for every quadratic equation. Memorise it!

Example: Solve 2x² - 4x - 3 = 0

a = 2, b = -4, c = -3
Discriminant: b² - 4ac = 16 + 24 = 40
x = (4 ± √40) / 4
x = (4 ± 2√10) / 4
x = (2 ± √10) / 2
x ≈ 2.58 or x ≈ -0.58

Method 3: Completing the Square

This method rewrites the equation in the form (x + p)² = q.

Example: Solve x² + 6x + 2 = 0

Move the constant: x² + 6x = -2
Half the coefficient of x, then square it: (6/2)² = 9
Add to both sides: x² + 6x + 9 = 7
Factor: (x + 3)² = 7
Solve: x + 3 = ±√7
x = -3 ± √7

The Discriminant — How Many Solutions?

The discriminant is Δ = b² - 4ac and tells you what kind of solutions to expect:

Δ > 0: Two distinct real roots
Δ = 0: One repeated real root (the parabola touches the x-axis)
Δ < 0: No real roots (the parabola doesn't cross the x-axis)

This is hugely important in matric maths — examiners love testing this!

Common Mistakes to Avoid

Forgetting to set the equation equal to zero before factoring or using the formula
Sign errors in the quadratic formula — be especially careful with negative b values
Not simplifying surds — always simplify your final answer
Dividing by x instead of factoring — this loses a root (x = 0)

Practice Problems

Try these yourself, then check your work:

x² - 7x + 12 = 0
3x² + 2x - 1 = 0
x² - 4x + 4 = 0
2x² + 5x + 3 = 0

Need More Help?

If you're struggling with quadratic equations or any other maths topic, I'm here to help. Book a session and we'll work through it together until it clicks.