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. A quadratic equation is any equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. Because the highest power of x is 2, the equation can have up to two solutions (called roots), and the shape of its graph is always a parabola. Depending on the numbers involved, some quadratics are quick to factor by inspection, others need the quadratic formula, and a few are cleanest when solved by completing the square. Knowing which method to reach for — and why it works — is what actually saves time in an exam, more than memorising the formula itself.
Below are 20 fully worked examples grouped by method, followed by an explanation of when to use each approach, the mistakes that cost students the most marks, and a short FAQ. Work through the examples with a pencil and paper rather than just reading them; quadratics are a "practice" topic, not a "read once" topic.
All three methods — factoring, the quadratic formula, and completing the square — solve exactly the same equation and will always agree on the final answer. They exist because each one is easier in different situations:
A practical exam strategy: glance at the equation first. If a = 1 and b, c are small integers, try factoring for 10–15 seconds. If nothing obvious appears, switch to the quadratic formula immediately rather than wasting time guessing factors.
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.
Which method should I use in an exam if I'm short on time?
Try factoring for a few seconds first. If a, b, c are simple integers and the numbers look like they could multiply out nicely, factor it. Otherwise go straight to the quadratic formula — it never fails, even when factoring would eventually work too.
What does it mean if the discriminant equals zero?
It means the quadratic has exactly one repeated real root, and the parabola just touches the x-axis at a single point instead of crossing it twice.
Do I need to memorise the quadratic formula, or can I derive it?
Most exams expect you to know it by heart, but understanding that it comes from completing the square on the general equation ax² + bx + c = 0 makes it much easier to recall under pressure — you can rebuild it even if your memory blanks.
Why do some quadratics have no real solutions?
Graphically, this happens when the parabola never crosses the x-axis at all — it stays entirely above or entirely below it. Algebraically, it shows up as a negative discriminant, which forces you to take the square root of a negative number.
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. Typing in the same a, b, and c values and comparing the calculator's steps to your own working is also one of the fastest ways to spot exactly where a mistake crept in.
Check your own answers step by step with the Quadratic Equation Solver. No sign-up needed. Also try the Fraction Calculator.