# Find the roots of the following quadratic equations by factorisation:

(i) x^{2} - 3x -10 = 0

(ii) 2x^{2} + x - 6 = 0

(iii) √2x^{2} + 7x + 5√2 = 0

(iv) 2x^{2} - x + 1/ 8 = 0

(v) 100x^{2} - 20x + 1= 0

**Solution:**

The roots of the polynomial are the same as the zeros of the polynomial.

Therefore, roots can be found by factorizing the quadratic equation into two linear factors and after that equating each factor to zero.

(I) x^{2} - 3x -10 = 0

x^{2} - 5x + 2x -10 = 0

x(x - 5) + 2(x - 5) = 0

(x - 5) (x + 2) = 0

x - 5 = 0 and x + 2 = 0

x = 5 and x = - 2

Therefore, roots are : - 2, 5

(ii) 2x^{2} + x - 6 = 0

2x^{2} + 4x - 3x - 6 = 0

2x (x + 2) - 3 (x + 2) = 0

(2x - 3) (x + 2) = 0

2x - 3 = 0 and x + 2 = 0

2x = 3 and x = - 2

x = 3/2 and x = - 2

Therefore, roots are: 3/2, -2

(iii) √2x^{2} + 7x + 5√2 = 0

√2x^{2} + 5x + 2x + 5√2 = 0

√2x^{2} + 2x + 5x + 5√2 = 0

(√2x + 5) (x + √2) = 0

√2x + 5 = 0 or x + √2 = 0

√2x = - 5 or x = - √2

x = - 5/√2 or x = - √2

Therefore, roots are: - 5/√2, - √2

(iv) 2x^{2} - x + 1/ 8 = 0

Multiplying both sides of the equation by 8:

2(8) x^{2} - 8(x) + (8)(1/ 8) = (0)8

16x^{2} - 8x + 1 = 0

16x^{2} - 4x - 4x + 1 = 0

4x (4x - 1) -1 (4x - 1) = 0

(4x - 1) (4x - 1) = 0

(4x - 1)^{2} = 0

4x - 1 = 0

x = 1/4 and x = 1/4

Roots are: 1/4, 1/4

(v) 100x^{2} - 20x + 1 = 0

100x^{2} - 20x + 1 = 0

100x^{2} - 10x - 10x + 1 = 0

10x(10x - 1) -1(10 x - 1) = 0

(10x - 1)(10 x - 1) = 0

(10x - 1)^{2} = 0

10x - 1 = 0

x =1/10 and x = 1/10

Roots are: 1/10, 1/10

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 4

**Video Solution:**

## Find the roots of the following quadratic equations by factorization: (i) x² - 3x -10 = 0 (ii) 2x² + x - 6 = 0 (iii) √2x² + 7x + 5√2 = 0 (iv) 2x² - x + 1/ 8 = 0 (v) 100x² - 20x + 1= 0

Class 10 Maths NCERT Solutions Chapter 4 Exercise 4.2 Question 1

**Summary:**

The roots of the quadratic equation by the method of the factorization for (i) x² - 3x -10 = 0 (ii) 2x² + x - 6 = 0 (iii) √2x² + 7x + 5√2 = 0 (iv) 2x² - x + 1/ 8 = 0 (v) 100x² - 20x + 1= 0 are (i) - 2, 5, (ii) 3/2, -2, (iii) - 5/√2, - √2, (iv) 1/4, 1/4 and (v) 1/10, 1/10 respectively.

**☛ Related Questions:**

- Check whether the followings are quadratic equations:i) (x +1)² = 2 ( x - 3)ii) x² - 2x = (-2) (3 - x)iii) (x - 2)( x +1) = (x -1)( x + 3)iv) (x - 3)(2x +1) = x ( x + 5)v) (2x -1)( x - 3) = ( x + 5)( x -1)vi) x² + 3x +1 = ( x - 2)²vii) (x + 2)³ = 2x (x² -1)viii) x³ - 4x²- x +1 = ( x - 2)³
- Represent the following situation in the form of quadratic equations:(i) The area of a rectangular plot is 528 m². The length of the plot (in meters) is one more than twice its breadth. We need to find the length and breadth of the plot.
- Solve the problems given in example 1.(i) John and Jivanti had 45 Both of them lost 5 marbles each and the product of the no. of marbles they now have is 124. We would like to find out how many marbles they had to start with?

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