Solving Quadratic Equations

Learn how to solve quadratic equations of the form ax² + bx + c = 0 by factorising, completing the square, and using the quadratic formula.

Learning Objectives

Understand the standard form of a quadratic equation ax² + bx + c = 0
Factorise quadratics into two linear brackets (x + p)(x + q) = 0
Apply the zero product property to find roots
Use the quadratic formula x = (-b ± √(b² - 4ac)) / (2a)
Recognise when a quadratic equation has 0, 1, or 2 real solutions

Concept Explanation

Quadratic equations are one of the most fundamental topics in GCSE and A-Level algebra. A quadratic equation contains an x² term as its highest power. Solving a quadratic means finding the value(s) of x that make the equation equal to zero.

A quadratic equation is any equation that can be rearranged into the standard form ax² + bx + c = 0, where a ≠ 0.

Method 1: Factorising. Find two numbers that multiply to give c (or a × c) and add together to give b. Then set each factor equal to zero.

Method 2: Quadratic Formula. Works for all quadratics, especially when the equation does not factorise neatly over integers.

The expression inside the square root, b² - 4ac, is called the Discriminant (Δ). If b² - 4ac > 0, there are 2 real roots. If b² - 4ac = 0, there is 1 repeated root. If b² - 4ac < 0, there are no real roots.

Key Mathematical Formulas

ax^2 + bx + c = 0

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

\Delta = b^2 - 4ac

Worked Example

Step-by-Step Solution

Interactive Walkthrough

Problem Statement:

Solve the quadratic equation by factorising:

x^2 + 5x + 6 = 0

Step 1: Identify numbers that multiply to 6 and add to 5

We need two numbers p and q such that p × q = +6 and p + q = +5. The factor pairs of 6 are (1, 6) and (2, 3). Testing sums: 2 + 3 = 5.

2 \times 3 = 6 \quad \text{and} \quad 2 + 3 = 5

Final Answer:

x = -2 \quad \text{or} \quad x = -3

Try It Yourself

Solve the quadratic equation x² - 7x + 12 = 0 by factorising. What are the two roots for x?

x^2 - 7x + 12 = 0

Common Student Mistakes

❌ Common Pitfall: Forgetting to flip signs when solving brackets

✅ Correction: If the bracket is (x + 3), setting (x + 3) = 0 gives x = -3. Do not assume the answer is +3.

❌ Common Pitfall: Forgetting the ± symbol when taking square roots

✅ Correction: If x² = 9, then x = +3 OR x = -3. Always write x = ±3.

❌ Common Pitfall: Dividing by x when x is a factor

✅ Correction: If x² + 5x = 0, do not divide by x! Factorise as x(x + 5) = 0 so you don't lose the root x = 0.

UK Exam Advice & Tips

Always check if the question specifies 'Give your answers to 2 decimal places' — if so, use the quadratic formula!
Rearrange your quadratic so it equals 0 before trying to factorise or use the formula.
Substitute your calculated x values back into the original equation to verify that both sides equal 0.

Lesson FAQ

Can every quadratic equation be factorised?

No, not all quadratics can be factorised using whole integers. If a quadratic does not factorise easily, you should use the quadratic formula or complete the square.

What happens if the discriminant b² - 4ac is negative?

If b² - 4ac < 0, the equation has no real solutions (no real roots) because you cannot take the square root of a negative number in real arithmetic. At A-Level Further Maths, this introduces complex numbers.

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