Quadratic equations: step-by-step guide, methods, examples, and practice
Use this page as the main quadratic hub on CureMath — AI Math Explainer: understand what makes an equation quadratic, compare the main solution methods, and then choose the right next page or tool.
Start here if you want the short version before reading the full method.
- A quadratic equation has a highest power of 2, often written in the form ax^2 + bx + c = 0.
- The three main solution routes are factorising, completing the square, and the quadratic formula.
- A good first check is whether the quadratic factors cleanly before you move to a more general method.
What this topic means and what to look for first.
Quadratic equations look similar on the page, but they do not always want the same method.
This guide is designed to help you recognise the structure first, then choose the fastest reliable route instead of forcing one method onto every example.
One reliable route through the topic.
- 1Rewrite the equation so one side is 0 and the terms are in descending powers.
- 2Check whether there is a common factor or an obvious factor pair that makes factorising the fastest route.
- 3If factorising is awkward, decide whether you want square form insight or a direct formula method.
- 4Solve carefully, keeping signs and coefficients visible at each step.
- 5Check the result by substitution, expansion, or by comparing the structure of the equation with the method used.
Choose the route that fits the quadratic.
Best when the quadratic breaks cleanly into brackets and you can spot the factor pair quickly.
Best when you want to understand the structure of the quadratic or connect the equation to its graph.
Best when the quadratic does not factor easily or you need one method that always works.
See the method in action.
x^2 + 5x + 6 = 0
- Look for two numbers that multiply to 6 and add to 5.
- Those numbers are 2 and 3, so the quadratic factorises to (x + 2)(x + 3) = 0.
- Set each bracket equal to 0 to get x = -2 or x = -3.
2x^2 + 7x + 3 = 0
- Use the ac method: 2 × 3 = 6, then find two numbers that multiply to 6 and add to 7.
- Split the middle term: 2x^2 + 6x + x + 3 = 0.
- Factor by grouping to get (2x + 1)(x + 3) = 0, so x = -1/2 or x = -3.
x^2 + 6x + 5 = 0
- Move the constant first: x^2 + 6x = -5.
- Half of 6 is 3, and 3^2 is 9, so add 9 to both sides: x^2 + 6x + 9 = 4.
- This becomes (x + 3)^2 = 4, so x + 3 = ±2 and the solutions are x = -1 or x = -5.
2x^2 + 3x - 2 = 0
- Identify a = 2, b = 3, and c = -2.
- Substitute into x = (-b ± √(b^2 - 4ac)) / 2a.
- This gives x = (-3 ± √25) / 4, so the solutions are x = 1/2 and x = -2.
Things that commonly send the method off track.
- Confusing a quadratic expression with a quadratic equation and forgetting that solving needs an equals sign.
- Choosing a factor pair that multiplies correctly but does not add to the middle coefficient.
- Dropping a negative sign when identifying a, b, or c for the quadratic formula.
- Not setting each factor equal to zero after factorising.
- Using a valid method but never checking whether the final roots actually satisfy the original equation.
Use a short verification pass before moving on.
- Substitute each root back into the original equation and confirm the result is 0.
- If you factorised, expand the brackets again to make sure they return to the original quadratic.
- If you used the quadratic formula, check the discriminant b^2 - 4ac to confirm whether the number of roots makes sense.
Try a few variations before switching to a calculator or solver tool.
- x^2 + 7x + 10 = 0
- x^2 - x - 12 = 0
- 2x^2 + 5x + 2 = 0
- x^2 + 4x + 1 = 0
- 3x^2 - 8x - 3 = 0
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Extra algebra revision resources
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Short answers worth checking.
It is an equation where the highest power of the variable is 2, often written as ax^2 + bx + c = 0.
Factorising is often the easiest when the brackets are obvious, but the best method depends on the structure of the quadratic.
Use it when the quadratic does not factor cleanly or when you want one general solving method that always works.
The discriminant b^2 - 4ac tells you: positive gives two real roots, zero gives one repeated root, and negative gives no real roots.
Continue with the next closely related topic.
Use the public site structure first, then switch into the solver tool only if you need a direct test.
CureMath uses artificial intelligence to suggest how a maths problem could potentially be solved. AI can make mistakes.
Check important answers independently before relying on them.