GCSE Maths · Maths
Quadratic equations: a clear study guide
A quadratic equation has a squared term, such as x². Choose a method that matches the form of the equation, and always check the solutions in the original equation.
Factorising simple quadratics
For x² + bx + c = 0, find two numbers that multiply to c and add to b. Write the expression as two brackets, set each bracket equal to zero, and solve both linear equations.
For example, x² + 5x + 6 = 0 becomes (x + 2)(x + 3) = 0, so x = −2 or x = −3.
The quadratic formula
For ax² + bx + c = 0, use x = (−b ± √(b² − 4ac)) ÷ 2a. Substitute a, b and c with their signs in brackets, and keep the plus-or-minus symbol until you calculate both answers.
The expression b² − 4ac is the discriminant. It helps indicate whether the equation has two, one or no real solutions.
Checking and presenting answers
Substitute each solution back into the original equation. If the question comes from a length, area or time problem, reject any solution that does not make sense in context and state the units where appropriate.
Quick check
What must a quadratic equation equal before using the formula?
Answer: Zero, in the form ax² + bx + c = 0.
Why are there sometimes two answers?
Answer: A quadratic graph can cross the x-axis at two points, giving two roots.
Common questions
Which method should I use for a quadratic equation?
Try factorising when the numbers are simple; use the formula when factorising is difficult or the question asks for an exact or decimal solution.
Why should I check my quadratic answers?
Checking catches sign and arithmetic errors and helps you reject solutions that are impossible in the original problem.