Quadratic Equations Class 10 Notes

This chapter is one of the highest weightage topics in CBSE Class 10 Maths. These notes are specially designed by Lakshya Institution, Gurugram to help students understand concepts deeply and score 90%+ in board exams.

Standard Form

Quadratic Formula

Discriminant

If D > 0 → Two distinct real roots
If D = 0 → Equal roots
If D < 0 → No real roots

Solved Examples (Board Pattern)

Example 1: Solve x² − 5x + 6 = 0

Solution:

Comparing with ax² + bx + c = 0

a = 1, b = -5, c = 6

D = b² − 4ac

D = (-5)² − 4(1)(6)

D = 25 − 24 = 1

x = [-(-5) ± √1] / 2(1)

x = [5 ± 1] / 2

x = 6/2 = 3

x = 4/2 = 2

Roots are 3 and 2.

Example 2: Find Nature of Roots of 2x² + 4x + 5 = 0

a = 2, b = 4, c = 5

D = b² − 4ac

D = 4² − 4(2)(5)

D = 16 − 40

D = -24

Since D < 0, equation has no real roots.

Previous Year Questions (PYQ)

PYQ 1 (CBSE Board)

Solve 3x² − 5x − 2 = 0 by quadratic formula.

Solution:

a = 3, b = -5, c = -2

D = (-5)² − 4(3)(-2)

D = 25 + 24 = 49

x = [5 ± √49] / 6

x = [5 ± 7] / 6

x = 12/6 = 2

x = -2/6 = -1/3

Roots are 2 and -1/3.

PYQ 2 (Word Problem)

The product of two consecutive positive integers is 132. Find the integers.

Solution:

Let first integer = x

Second integer = x + 1

x(x + 1) = 132

x² + x − 132 = 0

Factorizing:

x² + 12x − 11x − 132 = 0

x(x + 12) − 11(x + 12) = 0

(x − 11)(x + 12) = 0

x = 11 (positive integer)

Integers are 11 and 12.

PYQ 3 (Application Based)

The sum of squares of two consecutive natural numbers is 365. Find the numbers.

Let first number = x

Second number = x + 1

x² + (x + 1)² = 365

x² + x² + 2x + 1 = 365

2x² + 2x + 1 − 365 = 0

2x² + 2x − 364 = 0

Divide by 2:

x² + x − 182 = 0

Factorizing:

x² + 14x − 13x − 182 = 0

x(x + 14) − 13(x + 14) = 0

(x − 13)(x + 14) = 0

x = 13

Numbers are 13 and 14.

Important 4–5 Mark Board Questions

1. Find quadratic equation whose roots are 5 and -3.
2. Find value of k so that equation 2x² + kx + 8 = 0 has equal roots.
3. Find discriminant of x² − 4x + 4 and comment on nature of roots.
4. Solve: 4x² − 4x − 3 = 0
5. Find quadratic polynomial whose sum of roots is 3 and product is -10.

Exam Tips to Score Full Marks

• Always write formula before applying it.
• Calculate discriminant separately.
• Do not skip steps in 4-mark questions.
• Underline final answer.
• Check sign mistakes carefully.

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