Quadratic Equations – Class 10 Mathematics notes, important questions and MCQs

Chapter overview

A quadratic equation has the form ax² + bx + c = 0 (a ≠ 0). It is solved by factorisation or the quadratic formula, and the discriminant tells the nature of the roots.

Solving

• Factorisation: split the middle term so that the product equals ac.
• Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a.
• Roots are values of x that satisfy the equation.

Nature of roots (D = b² − 4ac)

• D > 0: two distinct real roots.
• D = 0: two equal real roots, x = −b/2a.
• D < 0: no real roots.

Word problems

• Form the equation from the situation (area, speed, age, numbers).
• Reject roots that do not make sense (negative length, negative speed).

Key terms

Quadratic equation

an equation of degree 2, ax² + bx + c = 0, a ≠ 0

Discriminant

D = b² − 4ac, decides the nature of the roots

Root

a value of x that makes the equation true

Formulas

• x = [−b ± √(b² − 4ac)] / 2a
• D = b² − 4ac

Important questions with answers

Quick revision

• ax² + bx + c = 0 with a ≠ 0.
• D > 0 two roots, D = 0 equal roots, D < 0 no real roots.
• Check whether both roots fit a word problem.

Exam tips

• Bring every term to one side (= 0) before factorising.
• For "equal roots" questions put D = 0 and solve for k.

Practice MCQs

1. Which is a quadratic equation?

   1. 2x² − 3x + 1 = 0
   2. x³ − x = 0
   3. (x + 2)² = x² + 4
   4. x + 1/x² = 3
   Show answer

   Answer: (a) 2x² − 3x + 1 = 0. (x + 2)² = x² + 4 reduces to 4x = 0 (linear); only 2x² − 3x + 1 = 0 has degree 2.

2. Roots of x² − 9 = 0 are

   1. ±9
   2. 3 only
   3. 9
   4. ±3
   Show answer

   Answer: (d) ±3. x² = 9 ⇒ x = ±3.

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

   1. 8
   2. 40
   3. −8
   4. 16
   Show answer

   Answer: (c) −8. 16 − 24 = −8.

4. Roots of x² − 5x + 6 = 0 are

   1. 2, 3
   2. −2, −3
   3. 1, 6
   4. −1, 6
   Show answer

   Answer: (a) 2, 3. (x − 2)(x − 3) = 0.

5. If 2x² + kx + 3 = 0 has equal roots, k =

   1. 24
   2. ±6
   3. ±2√6
   4. ±√6
   Show answer

   Answer: (c) ±2√6. k² − 24 = 0 ⇒ k = ±2√6.

6. x² + 4x + 5 = 0 has

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

   Answer: (b) no real roots. D = 16 − 20 < 0.

7. The product of two consecutive positive integers is 306. The smaller integer is

   1. 16
   2. 17
   3. 18
   4. 15
   Show answer

   Answer: (b) 17. x(x + 1) = 306 ⇒ x = 17.

8. A rectangle's length is 1 m more than twice its breadth and its area is 528 m². The breadth is

   1. 22 m
   2. 16 m
   3. 33 m
   4. 12 m
   Show answer

   Answer: (b) 16 m. x(2x + 1) = 528 ⇒ 2x² + x − 528 = 0 ⇒ x = 16.