--- title: "Difference of Two Squares and Perfect Square Trinomials" description: "Recognise the two quadratics that factorise on sight: a² - b² splitting into (a + b)(a - b), and a perfect square trinomial folding back into one bracket squared." canonical: https://lightmysky.com/learn/mathematics/difference-of-two-squares-and-perfect-square-trinomials-mt_GSpsHuserT source: https://lightmysky.com/learn/mathematics/difference-of-two-squares-and-perfect-square-trinomials-mt_GSpsHuserT.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`. # Difference of Two Squares and Perfect Square Trinomials Recognise the two quadratics that factorise on sight: a² - b² splitting into (a + b)(a - b), and a perfect square trinomial folding back into one bracket squared. Subject: Mathematics · Area: Algebra · Ages 14 to 15 Page: https://lightmysky.com/learn/mathematics/difference-of-two-squares-and-perfect-square-trinomials-mt_GSpsHuserT ## Ready when they can - Factorise x² - 49 and 9x² - 25 as a difference of two squares - Spot that x² + 10x + 25 is (x + 5)² from the halved middle coefficient - Use the difference of two squares to work out 51 × 49 mentally ## Lesson: Difference of Two Squares and Perfect Square Trinomials In the last stop you hunted for a pair that multiplies to c and adds to b. Two shapes skip the hunt entirely. One is a difference of two squares: two perfect squares joined by a minus. The other is a perfect square trinomial: one bracket, squared. Spotting them saves time, and they keep reappearing, from equations to mental maths. *(drawing: Why the pattern works. Expand (x + 7)(x - 7) and the -7x and +7x cancel, leaving only the two squares. That is why the middle term is missing.)* How to spot a difference of two squares: exactly two terms, each a perfect square, joined by a minus. x squared is a square and 49 is 7 squared, so x squared - 49 qualifies. But x squared + 49 does not, because the plus sign keeps the middle terms alive when you expand. **Example.** Factorise x squared - 49. The two squares are x squared and 7 squared, so the brackets are (x + 7)(x - 7). Check by expanding: the outer and inner terms give -7x and +7x, which cancel, and 7 times -7 gives -49. We land exactly on the original. **Example.** Now the x squared term carries a coefficient. Factorise 9x squared - 25. The first square is (3x) squared, since 3x times 3x gives 9x squared. The second square is 5 squared. So the brackets are (3x + 5)(3x - 5). Check: the middle terms -15x and +15x cancel, and 5 times -5 gives -25. **Example.** A perfect square trinomial folds back into one bracket squared. Expand (x + 5) squared: x squared, then 2 times 5x, then 25, which is x squared + 10x + 25. Running it backward, halve the middle coefficient: 10 over 2 gives 5. Check that 5 squared is 25, and the trinomial is (x + 5) squared. **Example.** The same identity does fast number work. 51 times 49 looks awkward, but 51 is 50 + 1 and 49 is 50 - 1. So 51 times 49 = (50 + 1)(50 - 1) = 50 squared - 1 squared = 2500 - 1 = 2499. One square, one subtraction, no long multiplication. **Recap.** A difference of two squares, a squared minus b squared, splits into (a + b)(a - b); a perfect square trinomial with middle term 2ab folds into (a + b) squared. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Factorising Quadratics into Two Brackets](https://lightmysky.com/learn/mathematics/factorising-quadratics-into-two-brackets-mt_3QXgmWKfoA) ## Opens up - [Factorising Quadratics with a Leading Coefficient](https://lightmysky.com/learn/mathematics/factorising-quadratics-with-a-leading-coefficient-mt_IdKGCzbH07) - [Completing the Square](https://lightmysky.com/learn/mathematics/completing-the-square-mt_rHmCfzB4Fm)