NCERT Class X Chapter 4: Quadratic Equation Example 4

NCERT Class X Chapter 4: Quadratic Equation Example 4

Question:

Find the roots of the quadratic equation 6x2 – x – 2 = 0.

Given:

The quadratic equation: 6x2 – x – 2 = 0

To Find:

The roots of the equation.

Formula:

For a quadratic equation ax² + bx + c = 0, we can find two numbers m and n such that:

                        ax² + bx + c = ax² + mx + nx + c

where

  • m × n = a × c
  • m + n = b

Cases:

  • If (a × c > 0) and b < 0, both m and n are negative.
  • If (a × c > 0) and b > 0, both m and n are positive.
  • If (a × c < 0), the numbers m and n have opposite signs.

Solution:

The given equation is 6x2 – x – 2 = 0.

Here, a = 6, b = -1, c = -2.
Product (a × c) = 6 × (-2) = -12.

Sum (b) = -1.
We need to find two numbers whose product is -12 and sum is -1.

Lets find the factor of 12.

We know that 12 = 4 x 3

⇒ 12 = 2 x 2 x 3

The factor of 12 are (2,6),(3,4),(4,3),(6,2)

Since product < 0 the numbers in the factors must be of opposite signs.

Therefore the possible (m,n) numbers are (-2,6),(-3,4),(-4,3) and (-6,2)

Lets calculate the sum,
                                          -2 + 6 = 4
                                          -3 + 4 = 1
                                          -4 + 3 = -1
                                          -6 + 2 = -4

Since the sum is -1 only for -4 and 3, the numbers are -4 and 3.
Rewrite the middle term (-x) using these numbers:

⇒ 6x2 – 4x + 3x – 2 = 0
Factor by grouping:

⇒ 2x(3x – 2) + 1(3x – 2) = 0
Factor out the common binomial (3x – 2):

⇒ (2x + 1)(3x – 2) = 0
Set each factor equal to zero to find the roots:

Case 1: 2x + 1 = 0

⇒ 2x = -1

⇒ x = -1 2
Case 2: 3x – 2 = 0

⇒ 3x = 2

⇒ x = 2 3

Result:

The roots of the equation 6x2 – x – 2 = 0 are x = -1 2 and x = 2 3 .
© Kaliyuga Ekalavya. All rights reserved.

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