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CBSE · Class 10 · Mathematics

Quadratic Equations

Standard difficulty · 20 questions · 40 marks

Name: Date: Score: ____ / 40

Show your working for Sections B and C. Answers are on the last page — try every question first!

Section A — Choose the correct answer8 × 1 = 8 marks
1.
The roots of x2 + 5x − 6 = 0 are:
  1. −6 and −1
  2. −1 and 6
  3. −6 and −5
  4. −6 and 1
2.
The discriminant of 5x2 + 5x + 5 = 0 is:
  1. 125
  2. 75
  3. −75
  4. −95
3.
The roots of x2 − 5x + 9 = 0 are:
  1. real and equal
  2. always zero
  3. real and distinct
  4. not real (no real roots)
4.
Which of these is a quadratic equation?
  1. 4x2 + x − 3 = 0
  2. (x + 1)2 = x2 − 3
  3. x3 + x − 3 = 0
  4. x − 3 = 0
5.
The standard form of a quadratic equation is:
  1. ax + b = 0
  2. ax2 + bx + c = 0, with a ≠ 0
  3. ax2 + bx + c = 0, with a = 0
  4. ax3 + bx2 + c = 0
6.
For what values of k does x2 + kx + 49 = 0 have equal roots?
  1. k = ±49
  2. k = ±7
  3. k = 14 only
  4. k = ±14
7.
One root of x2 − 14x + 45 = 0 is 9. The other root is:
  1. 5
  2. 14
  3. 45
  4. −5
8.
Assertion (A): x2 + x + 3 = 0 has two distinct real roots.
Reason (R): A quadratic equation has two distinct real roots when b2 − 4ac > 0.
  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true, but R is false.
  4. A is false, but R is true.
Section B — Short answer8 × 2 = 16 marks
9.
Solve by factorisation: 2x2 + 9x + 10 = 0
10.
Solve using the quadratic formula: 2x2 + 8x + 3 = 0
11.
Find the discriminant of 6x2 − 3x + 7 = 0 and say what kind of roots it has.
12.
Find k so that x2 + 14x + k = 0 has two equal roots.
13.
Solve: x + 18/x = 9 (x ≠ 0)
14.
Form the quadratic equation whose roots are 4 and −3.
15.
Solve: (x + 5)(x − 10) = −56
16.
For what values of k does x2 + kx + 9 = 0 have no real roots?
Section C — Word problems4 × 4 = 16 marks
17.
The product of two consecutive positive integers is 306. Find them.
18.
The length of a rectangular plot is 11 m more than its breadth. Its area is 900 m2. Find its length and breadth.
19.
The hypotenuse of a right-angled triangle is 159 cm. One of the other sides is 51 cm longer than the third. Find the two sides.
20.
A train travels 400 km at a uniform speed. If the speed were 20 km/h more, the journey would take 1 hour less. Find the speed of the train.
AmbifulWorksheets
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CBSE · Class 10 · Mathematics

Answer key

Quadratic Equations · AW-10M-QUA-2026-V1 · For parents — fold back or tear off before handing the sheet to your child.

QAnswerWorking
1(d) −6 and 1(x + 6)(x − 1) = 0
2(c) −75b2 − 4ac = 25 − 4 × 5 × 5 = −75
3(d) not real (no real roots)b2 − 4ac = −11 < 0
4(a) 4x2 + x − 3 = 0the highest power of x is 2, and the x2 term does not cancel
5(b) ax2 + bx + c = 0, with a ≠ 0—
6(d) k = ±14equal roots: k2 − 4 × 49 = 0, so k2 = 196
7(a) 5product of roots = 45, so the other is 45 ÷ 9 = 5
8(d) A is false, but R is true.A is false: b2 − 4ac = −11. R is true.
9x = −5/2 or x = −2(x + 2)(2x + 5) = 0
10x = (−4 ± √10)/2b2 − 4ac = 40; x = (−8 ± √40)/4
11−159; no real rootsb2 − 4ac = 9 − 168 = −159
12k = 49(14)2 − 4k = 0, so k = 49
13x = 3 or x = 6multiply by x: x2 − 9x + 18 = 0, so (x − 6)(x − 3) = 0
14x2 − x − 12 = 0x2 − (sum)x + product = 0: sum 1, product −12
15x = 2 or x = 3x2 − 5x − 50 = −56, so x2 − 5x + 6 = 0
16−6 < k < 6k2 − 36 < 0, so k2 < 36
1717 and 18x(x + 1) = 306; x2 + x − 306 = 0; x = 17
18length 36 m, breadth 25 mx(x + 11) = 900; x = 25
1984 cm and 135 cmx2 + (x + 51)2 = 25281; x = 84
2080 km/h400/x − 400/(x + 20) = 1; x2 + 20x − 8000 = 0; x = 80