The Quadratic Formula
The quadratic formula gives the solutions of any quadratic equation using its coefficients a, b and c.
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Play with the controls to see the idea in action.
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The quadratic formula gives the solutions of any quadratic equation using its coefficients a, b and c.
- x equals -b plus or minus the root of b squared minus 4ac all over 2a
- Works when factorising is difficult
- The discriminant decides the number of roots
🧠 Quick quiz
Three questions. Score 2 or more to complete the topic.
What is The Quadratic Formula?
Within Mathematics, which covers numbers, algebra, geometry, trigonometry and statistics from first principles to exam level, this topic explains The Quadratic Formula. Maths builds step by step, so each topic here gives you a rule, a picture and a quick check you can use on homework and in exams.
Check that you can explain The Quadratic Formula in your own words and give one example from everyday life. If you can do that, you have understood the core idea.
Understanding The Quadratic Formula gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
The Quadratic Formula in detail
The quadratic formula solves any equation of the form ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / 2a. The ± sign gives the two roots, one with plus and one with minus. It works even when the equation cannot be factorised easily.
Substitute carefully using brackets, especially when b is negative, as −b then becomes positive. Work out b² − 4ac first, then the square root. You can leave answers as surds or round them to a given accuracy. The formula comes from completing the square on the general quadratic.
How the interactive helps you understand The Quadratic Formula
Move the sliders to change the values in the equation. The curve, the moving point and, where they exist, the roots (the points where the curve crosses the x-axis) update straight away.
Before you change anything, predict the result. Then change one value at a time and check whether your prediction was right. This is a quick way to build a correct mental model of The Quadratic Formula.
Real-life uses and examples
- Use The Quadratic Formula when comparing prices, discounts or recipe quantities in everyday life.
- Explain The Quadratic Formula to a friend with a simple picture, such as a row of tiles or a bag of sweets.
- Spot The Quadratic Formula in exam questions, where it often sits inside a short word problem.
Common mistakes to avoid
- Forgetting units, signs or brackets when you apply The Quadratic Formula.
- Memorising the steps for The Quadratic Formula without checking why they work.
- Learning the definition without testing it. Use the quiz and the interactive to check that you can apply the idea.
Frequently asked questions about The Quadratic Formula
What is The Quadratic Formula?
The quadratic formula gives the solutions of any quadratic equation using its coefficients a, b and c.
What are the most important points about The Quadratic Formula?
The main points are: x equals -b plus or minus the root of b squared minus 4ac all over 2a; Works when factorising is difficult; The discriminant decides the number of roots.
Why is The Quadratic Formula useful?
Understanding The Quadratic Formula gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
How can I practise The Quadratic Formula?
Open the interactive at the top of this page and change the controls one at a time. Predict the result before each change, then take the three-question quiz. A score of two or more marks the topic as complete on this device.
Which topics are related to The Quadratic Formula?
Related topics in Mathematics include Inequalities, Equation of a Straight Line, Cubic and Reciprocal Graphs.