Keerthana Classes

Maths ch4 quadratic equations

October 04, 2026

37 marks. Read the notes, then tap Start test. Answers and your score appear only after you submit.

Quick notes

A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0.

A number α is a root if aα² + bα + c = 0. A quadratic equation has at most two roots.

Factorisation: split the middle term so that the two parts multiply to a × c and add to b, then factor and set each factor to zero.

Discriminant D = b² − 4ac.
D > 0 → two distinct real roots. D = 0 → two equal real roots. D < 0 → no real roots.

Part A: Multiple choice 16 questions

1. Which of these is a quadratic equation?

  1. x³ − 4x² = 1
  2. 2x + 5 = 0
  3. x² + 3x − 4 = 0
  4. x + 1/x = 3

The highest power of x is 2 and it is a polynomial equation.

2. In the standard form ax² + bx + c = 0, the condition needed is

  1. a ≠ 0
  2. b ≠ 0
  3. c ≠ 0
  4. a = 1

If a = 0 the x² term disappears and the equation is not quadratic.

3. The roots of x² − 3x − 10 = 0 are

  1. −5 and 2
  2. 5 and −2
  3. 10 and −1
  4. −10 and 1

x² − 3x − 10 = (x − 5)(x + 2).

4. The discriminant of 2x² − 4x + 3 = 0 is

  1. 8
  2. 40
  3. −8
  4. −40

D = (−4)² − 4 × 2 × 3 = 16 − 24 = −8.

5. If D < 0, the quadratic equation has

  1. two equal real roots
  2. two distinct real roots
  3. one real root
  4. no real roots

The square root of a negative number is not real.

6. If D = 0, the quadratic equation has

  1. two equal real roots
  2. no real roots
  3. two distinct real roots
  4. infinitely many roots

Both roots are −b/2a.

7. The nature of the roots of 2x² − 6x + 3 = 0 is

  1. no real roots
  2. two distinct real roots
  3. two equal real roots
  4. cannot be found

D = 36 − 24 = 12 > 0.

8. For 2x² + kx + 3 = 0 to have equal roots, k is

  1. ±6
  2. ±24
  3. ±2√6
  4. ±√6

k² − 24 = 0, so k = ±√24 = ±2√6.

9. The roots of 2x² + x − 6 = 0 are

  1. −3/2 and 2
  2. 3 and −2
  3. 1/2 and −3
  4. 3/2 and −2

2x² + 4x − 3x − 6 = (2x − 3)(x + 2).

10. Is (x − 2)² + 1 = 2x − 3 a quadratic equation?

  1. Yes
  2. No
  3. Only if x > 2
  4. Only if x = 2

It simplifies to x² − 6x + 8 = 0, which has degree 2.

11. The maximum number of roots of a quadratic equation is

  1. 1
  2. 2
  3. 3
  4. 4

A quadratic equation has at most two roots.

12. If x = 1 is a root of x² + kx + 5 = 0, then k is

  1. 6
  2. 5
  3. −6
  4. −5

1 + k + 5 = 0 gives k = −6.

13. The product of two consecutive positive integers is 306. The integers are

  1. 16 and 17
  2. 18 and 19
  3. 15 and 16
  4. 17 and 18

x(x + 1) = 306 gives (x − 17)(x + 18) = 0, so x = 17.

14. The roots of x² − 9 = 0 are

  1. 3 and −3
  2. 9 and −9
  3. 3 only
  4. no real roots

x² = 9, so x = ±3.

15. The quadratic equation whose roots are 2 and 3 is

  1. x² + 5x + 6 = 0
  2. x² − 5x + 6 = 0
  3. x² − 6x + 5 = 0
  4. x² + 5x − 6 = 0

(x − 2)(x − 3) = x² − 5x + 6.

16. The equation 3x² − 2x + 1/3 = 0 has

  1. no real roots
  2. two distinct real roots
  3. two equal real roots
  4. three roots

D = 4 − 4 × 3 × 1/3 = 0.

Part B: Fill in the blanks 8 blanks

17. The discriminant of ax² + bx + c = 0 is D = b² − [[4ac]].
18. If D > 0, the equation has two [[distinct|different|unequal]] real roots, and if D < 0 it has [[no|zero]] real roots.
19. x² − 5x + 6 = 0 factorises as (x − 2)(x − [[3]]) = 0.
20. The roots of x² − 16 = 0 are 4 and [[-4|−4]].
21. In a quadratic equation, the highest power of the variable is [[2|two]].
22. For kx(x − 2) + 6 = 0 to have two equal roots, k = [[6]].
23. The roots of 2x² − 7x + 3 = 0 are 3 and [[1/2|0.5]].

Part C: Questions and answers 9 questions

24. Find the roots of 2x² − x + 1/8 = 0 by factorisation.

Multiply by 8: 16x² − 8x + 1 = 0, i.e. (4x − 1)² = 0.
Roots: 1/4 and 1/4.

25. Find the roots of √2x² + 7x + 5√2 = 0 by factorisation.

Split 7x as 5x + 2x: √2x² + 5x + 2x + 5√2 = x(√2x + 5) + √2(√2x + 5) = (√2x + 5)(x + √2).
Roots: −5/√2 and −√2.

26. Find two numbers whose sum is 27 and product is 182.

Let the numbers be x and 27 − x. Then x(27 − x) = 182 → x² − 27x + 182 = 0 → (x − 13)(x − 14) = 0.
The numbers are 13 and 14.

27. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Let base = x, altitude = x − 7. Then x² + (x − 7)² = 169 → 2x² − 14x − 120 = 0 → x² − 7x − 60 = 0 → (x − 12)(x + 5) = 0.
x = 12 (length cannot be negative). Base 12 cm, altitude 5 cm.

28. A rectangular mango grove has length twice its breadth and area 800 m². Find its length and breadth.

Breadth = b, length = 2b. 2b² = 800 → b² = 400 → b = 20.
Breadth 20 m, length 40 m.

29. Find the nature of the roots of 3x² − 4√3x + 4 = 0, and find them if they are real.

D = (4√3)² − 4 × 3 × 4 = 48 − 48 = 0, so the roots are real and equal.
(√3x − 2)² = 0, so x = 2/√3 (both roots).

30. Is it possible to design a rectangular park of perimeter 80 m and area 400 m²? If so, find its length and breadth.

l + b = 40 and lb = 400. Let l = x: x(40 − x) = 400 → x² − 40x + 400 = 0 → (x − 20)² = 0.
Yes: length = breadth = 20 m (it is a square).

31. Is this situation possible? The sum of the ages of two friends is 20 years. Four years ago, the product of their ages was 48.

Ages x and 20 − x. (x − 4)(16 − x) = 48 → x² − 20x + 112 = 0.
D = 400 − 448 = −48 < 0, so there are no real roots. The situation is not possible.

32. A train travels 480 km at a uniform speed. If the speed were 8 km/h less, it would take 3 hours more. Find the speed of the train.

Let the speed be x km/h. 480/(x − 8) − 480/x = 3 → 480 × 8 = 3x(x − 8) → x² − 8x − 1280 = 0 → (x − 40)(x + 32) = 0.
Speed = 40 km/h.

Part D: Word puzzles 4 puzzles

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