NCERT Class X Chapter 4: Quadratic Equation Example 3

NCERT Class X Chapter 4: Quadratic Equation Example 3

Question:

Find the roots of the equation 2x2 – 5x + 3 = 0, by factorisation.

Given:

The quadratic equation: 2x2 – 5x + 3 = 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 – 5x + 3 = 0.

Here, a = 2, b = -5, c = 3.
Product (a × c) = 2 × 3 = 6.

Sum (b) = -5.
We need to find two numbers m and n whose product is 6 and sum is -5.

Lets find the factors of 6.

We know that, 2 x 3 = 6

Since product > 0 and sum < 0, m and n are negative

Therefore the two numbers are -2 and -3

Rewrite the middle term (-5x) using these numbers:

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

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

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

⇒ 2x = 3

⇒ x = 3 2
Case 2: x – 1 = 0

⇒ x = 1

Result:

The roots of the equation 2x2 – 5x + 3 = 0 are x = 3 2 and x = 1.
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