What Is a Quadratic Equation?
A quadratic equation is any equation of the form:
ax² + bx + c = 0, where a ≠ 0.
Quadratics appear throughout the O-Level Mathematics syllabus and are foundational for topics like graphs, inequalities, and simultaneous equations. Mastering them early pays dividends across the entire syllabus.
Method 1: Factorisation
Factorisation 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
- Find two numbers that multiply to 6 and add to 5 → 2 and 3
- Factorise: (x + 2)(x + 3) = 0
- Solve: x = −2 or x = −3
Tip: If you cannot find integer factors quickly, move on to another method.
Method 2: The Quadratic Formula
When factorisation isn't straightforward, use the quadratic formula:
x = (−b ± √(b² − 4ac)) / 2a
This formula always works and is especially useful when answers are decimals or surds.
Example
Solve: 2x² − 4x − 3 = 0
- Identify: a = 2, b = −4, c = −3
- Calculate the discriminant: b² − 4ac = 16 + 24 = 40
- Apply the formula: x = (4 ± √40) / 4
- Simplify: x ≈ 2.58 or x ≈ −0.58
Method 3: Completing the Square
Completing the square is essential for converting a quadratic to vertex form and is required knowledge for A-Maths. The steps are:
- Move the constant to the right: x² + bx = −c
- Add (b/2)² to both sides.
- Write the left side as a perfect square.
- Solve by taking the square root of both sides.
The Discriminant: How Many Solutions?
The expression b² − 4ac is called the discriminant and tells you how many real solutions a quadratic has:
| Discriminant Value | Number of Solutions |
|---|---|
| b² − 4ac > 0 | Two distinct real roots |
| b² − 4ac = 0 | One repeated real root |
| b² − 4ac < 0 | No real roots |
Common Mistakes to Avoid
- Forgetting to set the equation equal to zero before factorising.
- Sign errors when substituting negative values of b into the formula.
- Not simplifying surds fully in the final answer.
- Dividing through by x (which loses the solution x = 0).
Summary
Quadratic equations are a cornerstone of Secondary Maths. By mastering all three methods — factorisation, the quadratic formula, and completing the square — you'll be equipped to handle any variation that appears in your exam. Always check which method is most efficient for the question at hand.