A one-page summary of everything needed to factorise and solve quadratic equations: the sum-and-product method, common factors, difference of two squares, perfect squares, the null factor law and the quadratic formula. Each technique has a compact worked example.
Instant download · No sign-up · Free for personal, classroom and homeschool use
Curriculum: Australian Curriculum. We show specific outcome codes only where they have been verified against the official curriculum document.
Completed: Quadratics Cheat Sheet: Factorising and Solving
Year 10 students (typically ages 15–16) working on quadratics. It is pitched at level 3 · challenge.
This is a study guide, so there is no separate answer sheet; the 'Check yourself' questions include answers.
About 15 minutes. Short, regular sessions work best: two or three a week beats one long one.
No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.
Try Year 10 Factorising and Solving Quadratics Worksheet — Level 2, then check understanding with the Year 10 Quadratics Topic Test.
Year 10 · Mathematics · Quadratics
Student name:
Date:
A quadratic is an expression whose highest power is x2. A quadratic equation in standard form is ax[sup 2] + bx + c = 0. Solving it means finding the values of x that make it true; there can be two, one or no real solutions. The main method is to factorise and then use the null factor law: if two things multiply to give 0, at least one of them must be 0.
Expanding (the reverse of factorising). (x + 3)(x + 5) = x2 + 5x + 3x + 15 = x2 + 8x + 15. Notice that 3 + 5 = 8 (the middle coefficient) and 3 × 5 = 15 (the constant). Factorising runs this backwards.
Factorising x[sup 2] + bx + c. Find two numbers that multiply to give c and add to give b.
Three sum-and-product examples
| Special case | Pattern | Example |
|---|---|---|
| Common factor | Take out the HCF first | 3x2 + 6x = 3x(x + 2) |
| Difference of two squares | a2 − b2 = (a − b)(a + b) | x2 − 25 = (x − 5)(x + 5) |
| Perfect square | a2 + 2ab + b2 = (a + b)2 | x2 + 6x + 9 = (x + 3)2 |
Solve x2 − 2x − 15 = 0
Solve x2 = 5x
The quadratic formula. When a quadratic will not factorise easily, use x = [frac −b ± √(b[sup 2] − 4ac)/2a]. The part under the square root, b2 − 4ac, is the discriminant: positive means two solutions, zero means one solution, negative means no real solutions.
Solve 2x2 + 3x − 2 = 0
Answers: 1. (x + 4)(x + 5) 2. (x − 7)(x + 7) 3. (x + 5)(x − 2) = 0, so x = −5 or x = 2 4. (x − 3)2 = 0, so x = 3 5. 4x(x − 3) = 0, so x = 0 or x = 3 6. x = = −1 ± √2, so x ≈ 0.41 or x ≈ −2.41
Preview — the PDF contains the full resource.
Factorise monic quadratics and solve quadratic equations by factorising. Expected-level practice for this year group. This Year 10…
Factorise monic quadratics and solve quadratic equations by factorising. Expected-level practice for this year group. This Year 10…
Factorise monic quadratics and solve quadratic equations by factorising. A challenge set for students ready to stretch further. This Year…
Expanding binomials, factorising and solving quadratic equations in a 45-minute Year 10 test with marks and fully worked solutions.
Free to print and share for personal, classroom and homeschool use. Please don't resell. Spotted an error? Let us know.