# Class 10 RD Sharma Solutions – Chapter 8 Quadratic Equations – Exercise 8.4

• Last Updated : 30 Apr, 2021

### 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:

=> (ab)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

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