Quadratic Equations
A quadratic equation has the form ax squared plus bx plus c equals zero and can have two, one or no real solutions.
▶ Interact
Play with the controls to see the idea in action.
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A quadratic equation has the form ax squared plus bx plus c equals zero and can have two, one or no real solutions.
- Solve by factorising, formula or completing the square
- The graph is a parabola
- Roots are where the graph crosses the x-axis
🧠 Quick quiz
Three questions. Score 2 or more to complete the topic.
What is Quadratic Equations?
Within Mathematics, which covers numbers, algebra, geometry, trigonometry and statistics from first principles to exam level, this topic explains Quadratic Equations. 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 Quadratic Equations in your own words and give one example from everyday life. If you can do that, you have understood the core idea.
Understanding Quadratic Equations gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
Quadratic Equations in detail
A quadratic equation has the form ax² + bx + c = 0, where the highest power of x is 2. Its graph is a parabola and it can have two, one or no real solutions, called roots, where the curve meets the x-axis.
The simplest method is factorising: x² − 5x + 6 = 0 becomes (x − 2)(x − 3) = 0, so x = 2 or x = 3, because if two numbers multiply to zero one must be zero. Always rearrange the equation to equal zero first. Quadratics describe projectile motion, areas and optimisation problems.
How the interactive helps you understand Quadratic Equations
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 Quadratic Equations.
Real-life uses and examples
- Explain Quadratic Equations to a friend with a simple picture, such as a row of tiles or a bag of sweets.
- Spot Quadratic Equations in exam questions, where it often sits inside a short word problem.
- Check a homework answer about Quadratic Equations by estimating first, then working it out step by step.
Common mistakes to avoid
- Memorising the steps for Quadratic Equations without checking why they work.
- Forgetting units, signs or brackets when you apply Quadratic Equations.
- Learning the definition without testing it. Use the quiz and the interactive to check that you can apply the idea.
Frequently asked questions about Quadratic Equations
What is Quadratic Equations?
A quadratic equation has the form ax squared plus bx plus c equals zero and can have two, one or no real solutions.
What are the most important points about Quadratic Equations?
The main points are: Solve by factorising, formula or completing the square; The graph is a parabola; Roots are where the graph crosses the x-axis.
Why is Quadratic Equations useful?
Understanding Quadratic Equations gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
How can I practise Quadratic Equations?
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 Quadratic Equations?
Related topics in Mathematics include The Discriminant, Gradient of a Line, Quadratic Graphs.