GCSE Maths • Foundation & Higher
GCSE Quadratic Equations Revision Guide
Learn how to choose and apply the correct method for solving quadratics.
Foundation
Exam practice
Worked examples
What You'll Learn
Recognise a Quadratic
Identify equations and expressions whose highest power of the variable is two.
Solve by Factorising
Rewrite a quadratic as a product and apply the zero-product rule to find every solution.
Use the Quadratic Formula
Identify a, b and c correctly, substitute with brackets and calculate both possible roots.
Complete the Square
Rewrite a quadratic in completed-square form to solve equations or identify a turning point.
Interpret Quadratic Graphs
Connect roots, y-intercepts, symmetry and turning points with algebraic forms of a quadratic.
Check and Compare Methods
Substitute solutions, expand factors and use the discriminant to check the result and root type.
KEY RULES AND FORMULAE Rules
Quadratic Equation Lessons
Completing The Square
Rewrite quadratics in completed-square form, solve suitable equations and identify turning points from the new form.
Quadratic Formula
Apply the quadratic formula accurately, handle signed coefficients and interpret the value of the discriminant.
Solving Quadratic Equations By Factorising
Factorise a quadratic, use the zero-product rule and check each solution in the original equation.
Core Subtopic
Expanding Double Brackets
Multiply every term and collect like terms to check or reverse a quadratic factorisation.
Common-Factor Quadratics
Take out a shared factor first, including x, before using the zero-product rule.
Non-Monic Quadratics
Factorise expressions where the coefficient of x² is not 1 by checking both brackets carefully.
Roots and Intercepts
Read or calculate the x-values where a quadratic graph crosses or touches the x-axis.
Turning Points and Symmetry
Use a graph or completed-square form to identify the maximum or minimum and axis of symmetry.
Exact and Rounded Solutions
Leave answers in exact surd form when requested, or round only the final calculator value.
Worked Example
Choosing and Checking a Solving Method
QUESTION
Solve 2x² − 3x − 4 = 0 using the quadratic formula. Give each solution to 3 significant figures.
- Given information: The equation is already in ax² + bx + c = 0 form, so a = 2, b = −3 and c = −4.
- Method: Substitute the signed coefficients into the quadratic formula, simplify exactly and round only at the end.
- Write the formula: x = (−b ± √(b² − 4ac)) ÷ (2a).
- Substitute with brackets: x = (−(−3) ± √((−3)² − 4(2)(−4))) ÷ (2 × 2).
- Simplify the discriminant: (−3)² − 4(2)(−4) = 9 + 32 = 41.
- Write the exact solutions: x = (3 ± √41) ÷ 4.
- Calculate both values: x = 2.35078… or x = −0.850781… .
- Round to 3 significant figures: x = 2.35 or x = −0.851.
- x = 2.35 or x = −0.851, each to 3 significant figures.
- The predicted proportion is 195/600 = 0.325, matching the observed relative frequency.
- Do not use b = 3. The coefficient is b = −3, so brackets are essential when substituting and squaring.
- Common GCSE Mistakes
- Not Rearranging to Zero
- Losing a Negative Coefficient
- Dividing Only Part of the Numerator
- Forgetting the Second Root
- Using the Zero-Product Rule Too Early
- Rounding Before the Final Step
- Exam Tips
- Rearrange the equation into ax² + bx + c = 0 before deciding which method to use.
- Try factorising first when integer factors are easy to spot; otherwise use a permitted reliable method.
- When using the formula, write a, b and c separately with their signs before substituting.
- Show the factorisation, zero-product equations or full formula substitution so the method is clear.
- State both solutions and follow the requested form and accuracy, including exact surds where required.
Ready to Practice?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Quadratics Equation Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Continue Your GCSE Maths Revision
Frequently Asked Questions
A quadratic equation is an equation that can be written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable is 2. A quadratic expression does not include an equals sign, while a quadratic equation does.
First rearrange the equation so it equals zero. Factorising is usually quickest when suitable factors are easy to find. The quadratic formula works for every quadratic in standard form. Completing the square is useful when the question requests that form or asks about a turning point.
The graph of a quadratic may cross the x-axis twice, touch it once or miss it completely. Algebraically, this is determined by the discriminant b² − 4ac. A positive value gives two distinct real roots, zero gives one repeated real root and a negative value gives no real roots.
A root is an x-value for which the quadratic equals zero, so it corresponds to an x-intercept. A turning point is the maximum or minimum point of the parabola. Completed-square form shows the turning point directly, but it does not normally give the roots without further solving.
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