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:

=> (aâˆ’b)2 = a2âˆ’2Ã—aÃ—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:

=> (aâˆ’b)2=a2âˆ’2Ã—aÃ—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:

=> (aâˆ’b)2=a2âˆ’2Ã—aÃ—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:

=> (aâˆ’b)2=a2âˆ’2Ã—aÃ—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:

=> (aâˆ’b)2=a2âˆ’2Ã—aÃ—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+4âˆš3â€‹+3=0.

Solution:

Given: 4x2+4âˆš3â€‹+3=0

We have to make the equation a perfect square.

=> 4x2+4âˆš3â€‹+3=0

=>

=>

We know that,

=> (aâˆ’b)2=a2âˆ’2Ã—aÃ—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,

=> (aâˆ’b)2=a2âˆ’2Ã—aÃ—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,

=> (aâˆ’b)2=a2âˆ’2Ã—aÃ—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,

=> (aâˆ’b)2=a2âˆ’2Ã—aÃ—b+b2

Thus the equation can be written as:

=>

=>

=>

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

=>

=>  and

=> x = âˆš2 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

=> x2âˆ’2Ã—xÃ—2a+(2a)2âˆ’b2=0

We know that,

=> (aâˆ’b)2=a2âˆ’2Ã—aÃ—b+b2

Thus the equation can be written as:

=> x2âˆ’2Ã—2aÃ—x+(2a)2=b2

=> (x-2a)2 = b2

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

=> (x-2a) = Â±b

=> x= 2a+b and x = 2a-b

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