Quadratic Equations Class 10: Practice Questions
Solve Class 10 quadratic equations by factorisation and the quadratic formula. Includes discriminant checks, word problems and explained answers.
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. First put every term on one side and simplify. Try factorisation when the factors are easy to spot; otherwise use the quadratic formula. Check a word-problem answer against the context as well as the equation.
Choose a method
| Method | When it helps |
|---|---|
| Factorisation | You can find factor pairs that give the middle term. |
| Quadratic formula | Factor pairs are inconvenient or roots involve square roots. |
| Discriminant D = b² − 4ac | You need the number of real roots before solving. |
The quadratic formula is x = (−b ± √(b² − 4ac))/(2a). If D > 0 there are two distinct real roots; if D = 0 they are equal; if D < 0 there are no real roots.
Worked practice
1. Solve x² − 9x + 20 = 0
- 1(x − 4)(x − 5) = 0, so x = 4 or 5.
- 2Check x = 4: 16 − 36 + 20 = 0.
2. Solve 2x² + x − 6 = 0
- 1Split x as 4x − 3x: 2x(x + 2) − 3(x + 2) = 0.
- 2(2x − 3)(x + 2) = 0, so x = 3/2 or −2.
3. Solve x² − 4x + 1 = 0
- 1a = 1, b = −4, c = 1. D = 16 − 4 = 12.
- 2x = (4 ± √12)/2 = 2 ± √3. Keep the exact form unless a decimal is requested.
4. How many real roots does 3x² + 2x + 5 = 0 have?
- 1D = 2² − 4 × 3 × 5 = −56.
- 2There are no real roots. A negative discriminant does not mean the equation has degree less than two.
A rectangle word problem
5. A rectangle has area 48 cm² and is 2 cm longer than it is wide
- 1Let width be x cm. Then length is x + 2 cm and x(x + 2) = 48.
- 2x² + 2x − 48 = (x + 8)(x − 6) = 0.
- 3The algebraic roots are −8 and 6, but width must be positive. Width = 6 cm; length = 8 cm. Check area = 48 cm².
Try first, then check
- 6. x² + 5x + 6 = 0. Answer: −2 and −3.
- 7. 4x² − 12x + 9 = 0. Answer: x = 3/2, a repeated root since D = 0.
- 8. The product of two consecutive positive integers is 72. Answer: 8 and 9; n(n + 1) = 72 gives (n + 9)(n − 8) = 0.
Avoid these errors
The entire numerator, including the square-root term, is divided by 2a. Use parentheses when entering the formula into a calculator. Never discard a negative root automatically: exclude it only when the problem context demands a positive quantity, such as a length.