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

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

Question:

Find the roots of the following quadratic equations by factorisation : 2x2 – x + (1/8) = 0

Given:

The quadratic equation: 2x2 – x + 1 8 = 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 + 1 8 = 0.
First, clear the fraction by multiplying the entire equation by 8:
⇒ 8(2x2) – 8(x) + 8( 1 8 ) = 8(0)
⇒ 16x2 – 8x + 1 = 0

Here, a = 16, b = -8, c = 1.
Product (a × c) = 16 × 1 = 16.

Sum (b) = -8.
We need to find two numbers whose product is 16 and sum is -8.

WKT, 16 = 4 x 4

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

Since product is 16 and 16 > 0 And Since sum is -8 and -8 < 0 both numbers are negative.

Therefore the two numbers are -4 and -4.
Rewrite the middle term (-8x) using these numbers:

⇒ 16x2 – 4x – 4x + 1 = 0
Factor by grouping:

⇒ 4x(4x – 1) – 1(4x – 1) = 0
Factor out the common binomial (4x – 1):

⇒ (4x – 1)(4x – 1) = 0
This is a perfect square trinomial, (4x – 1)2 = 0.
Set the factor equal to zero to find the root:

⇒ 4x – 1 = 0

⇒ 4x = 1

⇒ x = 1 4

Result:

The roots of the equation 2x2 – x + 1 8 = 0 are x = 1 4 (repeated root).
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