A one-page summary of everything needed to factor and solve quadratic equations: the sum-and-product method, common factors, difference of two squares, perfect squares, the zero product property and the quadratic formula. Each technique has a compact worked example.
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Curriculum: Texas grade levels (TEKS). We show specific outcome codes only where they have been verified against the official curriculum document.
Completed: Quadratics Cheat Sheet: Factoring and Solving
10th Grade 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.
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Try 10th Grade Factoring and Solving Quadratics Worksheet — Level 1, then check understanding with the 10th Grade Quadratics Topic Test.
Success TutoringWorksheets10th Grade · Mathematics · QuadraticsA 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 factor and then use the zero product property: if two things multiply to give 0, at least one of them must be 0.
Expanding (the reverse of factoring). (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). Factoring runs this backwards.
Factoring 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 factor 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
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