--- title: "Factorising Quadratics into Two Brackets" description: "Undo the expansion of two brackets for quadratics of the form x² + bx + c by finding the number pair that multiplies to c and adds to b. Check every answer by expanding it back." canonical: https://lightmysky.com/learn/mathematics/factorising-quadratics-into-two-brackets-mt_3QXgmWKfoA source: https://lightmysky.com/learn/mathematics/factorising-quadratics-into-two-brackets-mt_3QXgmWKfoA.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`. # Factorising Quadratics into Two Brackets Undo the expansion of two brackets for quadratics of the form x² + bx + c by finding the number pair that multiplies to c and adds to b. Check every answer by expanding it back. Subject: Mathematics · Area: Algebra · Ages 14 to 15 Page: https://lightmysky.com/learn/mathematics/factorising-quadratics-into-two-brackets-mt_3QXgmWKfoA ## Ready when they can - Factorise x² + 7x + 12 by finding the pair that multiplies to 12 and adds to 7 - Choose signs correctly for a quadratic with a negative constant, such as x² - 2x - 15 - Check a factorisation by expanding the brackets back to the original quadratic ## Lesson: Factorising Quadratics into Two Brackets Expanding double brackets turned (x + 3)(x + 5) into x squared + 8x + 15. Factorising runs that same move in reverse. You are handed the expanded answer and have to find the two brackets that made it. The skill pays off right away, because factorised form is the key that opens quadratic equations, which are next on the thread. *(drawing: The five moves of the method, in order. One pair of numbers does both jobs at once: it multiplies to c and adds to b.)* Here is the rule that does the work. The pair always multiplies to c and adds to b. The signs fall out of that. If c is positive, both numbers in the pair carry the sign of b. If c is negative, the pair has different signs, and the bigger number takes the sign of b. **Example.** Factorise x squared + 8x + 15. Here b is 8 and c is 15, so the pair is two positive numbers. The factor pairs of 15 are 3 and 5, or 1 and 15. Only 3 and 5 adds to 8, so the brackets are (x + 3)(x + 5). Now the check: expand back. x times x gives x squared, the outer and inner terms give 5x and 3x, and 3 times 5 gives 15. The x terms collect to 8x, so we land exactly on the original. **Example.** Now a negative constant. Factorise x squared + 2x - 15. Here b is 2 and c is -15, so the pair has different signs. The candidates are 5 and -3, or 3 and -5. Only 5 and -3 adds to 2, so the brackets are (x + 5)(x - 3). Check by expanding: the outer and inner terms give 5x and -3x, which collect to 2x, and 5 times -3 gives -15. We land back on the original. **Tip.** The order of the brackets never matters. (x + 3)(x + 5) and (x + 5)(x + 3) are the same answer, because multiplying has no order. Write whichever pair feels natural. Two habits to keep. If no pair of integers does both jobs, the quadratic cannot be factorised over the integers, and saying so is the correct answer. And make the check automatic: expand the brackets back and confirm you land on the original every time. **Recap.** Find the pair that multiplies to c and adds to b, put it in brackets with x, then check by expanding back out. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Expanding Double Brackets](https://lightmysky.com/learn/mathematics/expanding-double-brackets-mt_KhS7K1Mgrw) - [Factorising Expressions](https://lightmysky.com/learn/mathematics/factorising-expressions-mt_uESbzWCZIq) ## Opens up - [Difference of Two Squares and Perfect Square Trinomials](https://lightmysky.com/learn/mathematics/difference-of-two-squares-and-perfect-square-trinomials-mt_GSpsHuserT)