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

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

Question:

Find the roots of the following quadratic equations by factorisation: x2 – 3x – 10 = 0

Given:

The quadratic equation: x2 – 3x – 10 = 0

To Find:

The roots of the equation by factorization.

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 x2 – 3x – 10 = 0.

Here, a = 1, b = -3, c = -10.
Product (a × c) = 1 × (-10) = -10.

Sum (b) = -3.
We need to find two numbers whose product is -10 and sum is -3.

Take the factors of the product without sign.

Since product is -10, take 10.

WKT, 10 = 5 x 2

The two numbers we want are m = 5 and n = 2.

Since product is -10 and -10 < both numbers must have opposite sign.

Case 1: Let us consider -2 and 5. Product is -10 but sum is 3. So, the numbers cannot be -2 and 5

Case 2: Let us consider 2 and -5. Product is -10 and the sum is -3.

Therefore the two numbers are -5 and 2.
Rewrite the middle term (-3x) using these numbers:

⇒ x2 – 5x + 2x – 10 = 0
Factor by grouping:

⇒ x(x – 5) + 2(x – 5) = 0
Factor out the common binomial (x – 5):

⇒ (x + 2)(x – 5) = 0
Set each factor equal to zero to find the roots:

Case 1: x + 2 = 0

⇒ x = -2
Case 2: x – 5 = 0

⇒ x = 5

Result:

The roots of the equation x2 – 3x – 10 = 0 are x = -2 and x = 5.
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