--- title: "Solving Quadratic Equations by Factorising" description: "Rearrange a quadratic equation so one side is zero, factorise it, then use the fact that a product is zero only when one factor is zero to read off both solutions." canonical: https://lightmysky.com/learn/mathematics/solving-quadratic-equations-by-factorising-mt_tg_sqRzDjU source: https://lightmysky.com/learn/mathematics/solving-quadratic-equations-by-factorising-mt_tg_sqRzDjU.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Solving Quadratic Equations by Factorising Rearrange a quadratic equation so one side is zero, factorise it, then use the fact that a product is zero only when one factor is zero to read off both solutions. Subject: Mathematics · Area: Algebra · Ages 14 to 15 Page: https://lightmysky.com/learn/mathematics/solving-quadratic-equations-by-factorising-mt_tg_sqRzDjU ## Ready when they can - Rearrange x² + 5x = 14 into x² + 5x - 14 = 0 before factorising - Apply the zero product rule to get both solutions from (x - 2)(x + 7) = 0 - Reject a solution that cannot fit the context, such as a negative length ## Lesson: Solving Quadratic Equations by Factorising The last three stops gave you a factorising toolkit: the pair method, the special patterns, and splitting the middle term. This stop turns factorising into a way to solve equations. The key is a small fact: a product is zero only when one of its factors is zero. If you can write the left side as two brackets, each bracket hands you a solution. *(drawing: The moves of solving by factorising, in the order they happen.)* Here is the rule in full. If A times B equals zero, then A is zero or B is zero. There is no other way for a product to be zero. The common slip is to set the two factors equal to each other, like writing x - 3 = x + 4 from (x - 3)(x + 4) = 0. The equation says the product is zero, so each factor gets its own equation: x - 3 = 0 and x + 4 = 0. **Example.** Solve x squared + 3x - 10 = 0. The equation is already set to zero, so factorise straight away. The pair that multiplies to -10 and adds to 3 is 5 and -2, so the left side is (x + 5)(x - 2). Now the zero product rule: x + 5 = 0 or x - 2 = 0, giving x = -5 or x = 2. Check both in the original: (-5) squared + 3(-5) - 10 is 25 - 15 - 10 = 0, and 2 squared + 3(2) - 10 is 4 + 6 - 10 = 0. **Example.** Now an equation that is not yet set to zero. Solve 3x squared = 12x. Move every term to one side: 3x squared - 12x = 0. Take out the common factor 3x: 3x(x - 4) = 0. The rule gives x = 0 or x = 4. Notice why you must not divide both sides by x: dividing quietly drops the solution x = 0, because it assumes x is not zero. **Example.** Factorising also works when the answer has to make sense. A garden has width x metres and length (x + 3) metres, and its area is 40 square metres. So x(x + 3) = 40. Expand and set to zero: x squared + 3x - 40 = 0. The pair that multiplies to -40 and adds to 3 is 8 and -5, so (x + 8)(x - 5) = 0, giving x = -8 or x = 5. A width of -8 metres is impossible, so x = 5: the garden is 5 by 8 metres. **Tip.** The two solutions can look very different: one positive and one negative, one a fraction and one a whole number. That is normal. Trust the rule, write down both, and only then check them against the original equation or the context. **Recap.** Move every term to one side so the other is zero, factorise, set each factor to zero, and keep the solutions that fit the context. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Factorising Quadratics with a Leading Coefficient](https://lightmysky.com/learn/mathematics/factorising-quadratics-with-a-leading-coefficient-mt_IdKGCzbH07) ## Opens up - [Simplifying Algebraic Fractions](https://lightmysky.com/learn/mathematics/simplifying-algebraic-fractions-mt_scDNSoz3eW) - [Polynomial Division and the Factor Theorem](https://lightmysky.com/learn/mathematics/polynomial-division-and-the-factor-theorem-mt_haL2UkGa5C)