Quadratic equations show up in almost every FSc/Matric math paper, and most students lose marks not because they don't know the formula, but because they rush the steps. Below are 20 worked examples grouped by method, so you can see exactly how each type is solved.
1. Solve x² − 5x + 6 = 0.
Factor: (x − 2)(x − 3) = 0 → x = 2 or x = 3.
2. Solve x² + 7x + 12 = 0.
Factor: (x + 3)(x + 4) = 0 → x = −3 or x = −4.
3. Solve x² − 9 = 0.
Difference of squares: (x − 3)(x + 3) = 0 → x = 3 or x = −3.
4. Solve 2x² − 3x − 2 = 0.
Factor: (2x + 1)(x − 2) = 0 → x = −1/2 or x = 2.
5. Solve x² − x − 6 = 0.
Factor: (x − 3)(x + 2) = 0 → x = 3 or x = −2.
6. Solve x² + 4x = 0.
Factor out x: x(x + 4) = 0 → x = 0 or x = −4.
7. Solve 3x² − 12 = 0.
Divide by 3: x² − 4 = 0 → (x−2)(x+2) = 0 → x = 2 or x = −2.
Recall: for ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / 2a.
8. Solve x² + 2x − 5 = 0.
a=1, b=2, c=−5. Discriminant = 4+20 = 24. x = (−2 ± √24)/2 ≈ 1.45 or −3.45.
9. Solve 2x² − 4x + 1 = 0.
a=2, b=−4, c=1. Discriminant = 16−8 = 8. x = (4 ± √8)/4 ≈ 1.71 or 0.29.
10. Solve x² − 6x + 9 = 0.
Discriminant = 36−36 = 0 → one repeated root: x = 3.
11. Solve 3x² + 5x − 2 = 0.
Discriminant = 25+24 = 49. x = (−5 ± 7)/6 → x = 1/3 or x = −2.
12. Solve x² + x + 1 = 0.
Discriminant = 1−4 = −3 (negative) → no real roots (complex roots only).
13. Solve 4x² − 4x − 3 = 0.
Discriminant = 16+48 = 64. x = (4 ± 8)/8 → x = 3/2 or x = −1/2.
14. Solve x² − 3x − 1 = 0.
Discriminant = 9+4 = 13. x = (3 ± √13)/2 ≈ 3.30 or −0.30.
15. Solve x² + 6x + 5 = 0.
x² + 6x = −5 → x² + 6x + 9 = 4 → (x+3)² = 4 → x+3 = ±2 → x = −1 or −5.
16. Solve x² − 4x − 1 = 0.
x² − 4x = 1 → (x−2)² = 5 → x = 2 ± √5.
17. Solve x² + 2x − 8 = 0.
(x+1)² = 9 → x+1 = ±3 → x = 2 or x = −4.
18. Solve x² − 8x + 12 = 0.
(x−4)² = 4 → x−4 = ±2 → x = 6 or x = 2.
19. A rectangle's length is 3 more than its width, and its area is 40 m². Find the width.
Let width = x, length = x+3. x(x+3) = 40 → x² + 3x − 40 = 0 → (x+8)(x−5)=0 → x = 5 (width can't be negative).
20. The product of two consecutive positive integers is 132. Find the integers.
Let integers be x and x+1. x(x+1) = 132 → x² + x − 132 = 0 → (x−11)(x+12)=0 → x = 11. The integers are 11 and 12.
Once you've worked through a problem by hand, verify your answer using Calvo's Equation Solver — it's a good way to catch small arithmetic slips before an exam.