NCERT Class X Chapter 4: Quadratic Equation Exercise 4.2 Question 1 (ii)

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.2 Question 1 (ii)

Question:

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

Given:

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

To Find:

The roots of the equation by factorisation.

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 2x2 + x – 6 = 0.

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

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

Since product is -12, take 12.

WKT, 12 = 4 x 3

The two numbers we want are m = 4 and n = 3.

Since product is -12 and -12 < both numbers must have opposite sign.

Case 1:

Let us consider -4 and 3.

Product = -12 and  sum = -1.

So, the numbers cannot be -4 and 3

Case 2:

Let us consider 4 and -3.

Product = -12 and sum = 1.

Therefore the two numbers are 4 and -3.
Rewrite the middle term (x) using these numbers:

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

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

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

Case 1:  2x – 3 = 0

⇒ 2x = 3

⇒ x = 3 2
Case 2:  x + 2 = 0

⇒ x = -2

Result:

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

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