Class 10 RD Sharma Solutions - Chapter 8 Quadratic Equations - Exercise 8.4

Question 1: Find the roots of the following quadratic (if they exist) by the method of completing the square: .

Solution:

Given: 

We have to make the equation a perfect square.

=> 

=> 

We know that:

=> (a−b)² = a²−2×a×b+b²

Thus, the equation can be written as:

=> 

=> 

=> 

=> 

=> 

The RHS is positive, which implies that the roots exist.

=> 

=> x =  and x= 

=> x =  and x = 

Question 2: Find the roots of the following quadratic (if they exist) by the method of completing the square: 2x²-7x+3 = 0.

Solution:

Given: 2x²-7x+3 = 0

We have to make the equation a perfect square.

=> 2x²-7x+3 = 0

=> 

=> 

We know that:

=> (a−b)²=a²−2×a×b+b²

Thus, the equation can be written as:

=> 

=> 

=> 

The RHS is positive, which implies that the roots exist.

=> 

=>  and 

=>  and 

=> x = 3 and 

Question 3: Find the roots of the following quadratic (if they exist) by the method of the completing the square: 3x²+11x+10 = 0.

Solution:

Given: 3x²+11x+10 = 0

We have to make the equation a perfect square.

=> 3x²+11x+10 = 0

=> 

=> 

We know that:

=> (a−b)²=a²−2×a×b+b²

Thus the equation can be written as:

=> 

=> 

The RHS is positive, which implies that the roots exist.

=> 

=>  and 

=>  and 

=>  and x = -2

Question 4: Find the roots of the following quadratic (if they exist) by the method of completing the square: 2x²+x-4 =0.

Solution:

Given: 2x²+x-4 =0

We have to make the equation a perfect square.

=> 2x²+x-4 =0

=> 

=> 

We know that:

=> (a−b)²=a²−2×a×b+b²

Thus the equation can be written as:

=> 

The RHS is positive, which implies that the roots exist.

=>

=>  and 

Question 5: Find the roots of the following quadratic (if they exist) by the method of completing the square: 2x²+x+4 =0.

Solution:

Given: 2x²+x+4 =0

We have to make the equation a perfect square.

=> 2x²+x+4 =0

=> 

=> 

We know that:

=> (a−b)²=a²−2×a×b+b²

Thus the equation can be written as:

=> 

=> The RHS is negative, which implies that the roots are not real.

Question 6: Find the roots of the following quadratic (if they exist) by the method of completing the square: 4x²+4√3​+3=0.

Solution:

Given: 4x²+4√3​+3=0

We have to make the equation a perfect square.

=> 4x²+4√3​+3=0

=> 

=> 

We know that,

=> (a−b)²=a²−2×a×b+b²

Thus the equation can be written as:

=> 

=> 

=> 

The RHS is zero, which implies that the roots exist and are equal.

=> 

Question 7: Find the roots of the following quadratic (if they exist) by the method of the completing the square: .

Solution:

Given: 

We have to make the equation a perfect square.

=> 

=> 

=> 

We know that,

=> (a−b)²=a²−2×a×b+b²

Thus the equation can be written as:

=> 

=> 

=> 

The RHS is positive, which implies that the roots exist.

=> 

=>  and 

=>  and 

Question 8: Find the roots of the following quadratic (if they exist) by the method of completing the square: .

Solution:

Given: 

We have to make the equation a perfect square.

=> 

=> 

=> 

We know that,

=> (a−b)²=a²−2×a×b+b²

Thus the equation can be written as:

=> 

=> 

=> 

The RHS is positive, which implies that the roots exist.

=> 

=>  and 

=>  and 

Question 9: Find the roots of the following quadratic (if they exist) by the method of completing the square: .

Solution:

Given: 

We have to make the equation a perfect square.

=> 

=> 

We know that,

=> (a−b)²=a²−2×a×b+b²

Thus the equation can be written as:

=> 

=> 

=> 

The RHS is positive, which implies that the roots exist.

=> 

=>  and 

=> x = √2 and x = 1

Question 10: Find the roots of the following quadratic equation (if they exist) by the method of completing the square: x²-4ax+4a²-b²=0.

Solution:

Given: x²-4ax+4a²-b²=0

We have to make the equation a perfect square.

=> x²-4ax+4a²-b²=0

=> x²−2×x×2a+(2a)²−b²=0

We know that,

=> (a−b)²=a²−2×a×b+b²

Thus the equation can be written as:

=> x²−2×2a×x+(2a)²=b²

=> (x-2a)² = b²

The RHS is positive, which implies that the roots exist.

=> (x-2a) = ±b

=> x= 2a+b and x = 2a-b