Class 10 RD Sharma Solutions – Chapter 8 Quadratic Equations – Exercise 8.5

Question 1. Find the discriminant of the following quadratic equations:

(i) 2x² – 5x + 3 = 0  

Solution:

Given quadratic equation: 2x² – 5x + 3 = 0   ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0   ….(2)

On comparing eq(1) and (2), we get

Here, a = 2, b = -5 and c = 3

Now, we find the discriminant(D) = b² – 4ac

D = (-5)² – 4(2)(3)

= 25 – 24

= 1

Hence, the discriminant of given quadratic equation is 1

(ii) x² + 2x + 4 = 0

Solution:

Given quadratic equation: x² + 2x + 4 = 0   ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0   ….(2)

On comparing eq(1) and (2), we get

Here, a = 1, b = 2 and c = 4

Now, we find the discriminant(D) = b² – 4ac

D = (2)² – 4(1)(4)

= 4 – 16

= -12

Hence, the discriminant of given quadratic equation is -12

(iii) (x – 1) (2x – 1)

Solution:

Given quadratic equation:(x – 1)(2x – 1)

Or we can also write as, 2x² – 3x + 1 = 0   ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0   ….(2)

On comparing eq(1) and (2), we get

Here, a = 2, b = -3 and c = 1

Now, we find the discriminant(D) = b² – 4ac

D = (-3)² – 4(2)(1)

= 9 – 8

= 1

Hence, the discriminant of given quadratic equation is 1

(iv) x² – 2x + k = 0

Solution:

Given quadratic equation: x² – 2x + k = 0   ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0   ….(2)

On comparing eq(1) and (2), we get

Here, a = 1, b = -2 and c = k

Now, we find the discriminant(D) = b² – 4ac

D = (-2)² – 4(1)(k)

= 4 -4k

Hence, the discriminant of given quadratic equation is 4 – 4k

(v) √3x² + 2√2x – 2√3 = 0 

Solution:

Given quadratic equation:√3x² + 2√2x – 2√3 = 0    ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0   ….(2)

On comparing eq(1) and (2), we get

Here, a =√3, b = 2√2, and c = -2 – 2√3

Now, we find the discriminant(D) = b² – 4ac

D = (2√2)² – 4√3(-2√3)

= 8 + 24

= 32

Hence, the discriminant of given quadratic equation is 32

(vi) x² – x + 1 = 0

Solution:

Given quadratic equation: x² – x + 1 = 0   ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0   ….(2)

On comparing eq(1) and (2), we get

Here, a = 1, b = -1 and c = 1

Now, we find the discriminant(D) = b² – 4ac

D = (-1)² – 4(1)(1)

= 1 – 4

= -3

Hence, the discriminant of given quadratic equation is -3

Question 2. In the following, determine whether the given quadratic equation have real roots and If so, find the roots:

(i) 16x² = 24x + 1

Solution:

Given quadratic equation: 16x² – 24x – 1 = 0   ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 16, b = -24, and c = -1

Now, we find the discriminant(D) = b² – 4ac

D = (-24)² – 4(16)(-1)

= 576 + 64

= 640

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

(ii) x² + x + 2 = 0

Solution:

Given quadratic equation: x² + x + 2 = 0   ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 1, b = 1 and c = 2

Now, we find the discriminant(D) = b² – 4ac

D = (1)² – 4(1)(2)

= 1 – 8

= -7

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation does not satisfies the given condition, so it does not have real roots.

(iii) √3x² + 10x – 8√3 = 0

Solution:

Given quadratic equation: √3x² + 10x – 8√3 = 0   ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = √3, b = 10 and c = -8√3 

Now, we find the discriminant(D) = b² – 4ac

D = (10)² – 4(√3)(-8√3)

= 100 + 96

= 196

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

(iv) 3x² – 2x + 2 = 0

Solution:

Given quadratic equation: 3x² – 2x + 2 = 0   ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 3, b = -2 and c = 2

Now, we find the discriminant(D) = b² – 4ac

D = (-2)² – 4(3)(2)

= 4 – 24

= -20

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation does not satisfy the given condition, so it does not have real roots.

(v) 2x² – 2√6x + 3 = 0

Solution:

Given quadratic equation: 2x² – 2√6x + 3 = 0  ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 2, b= -2√6 and c = 3

Now, we find the discriminant(D) = b² – 4ac

D = (-2√6)² – 4(2)(3)

= 24 – 24

= 0

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

(vi) 3a²x² + 8abx + 4b² = 0 

Solution:

Given quadratic equation: 3a²x² + 8abx + 4b² = 0   ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 3a², b = 8ab and c = 4b²

Now, we find the discriminant(D) = b² – 4ac

D = (8ab)² – 4(3a²)(4b²)

= 64a²b² – 48a²b²

= 16a²b²

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

(vii) 3x² – 2√5x – 5 = 0 

Solution:

Given quadratic equation: 3x² – 2√5x – 5 = 0    ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 3, b = 2√5 and c = -5

Now, we find the discriminant(D) = b² – 4ac

D = (2√5 )² – 4(3)(-5)

= 20 + 60 

= 80

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

x = √5 /3

x = -√5 

(viii) x² – 2x + 1 = 0

Solution:

Given quadratic equation: x² – 2x + 1 = 0    ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 1, b = -2 and c = 1

Now, we find the discriminant(D) = b² – 4ac

D = (-2)² – 4(1)(1)

= 4 – 4

= 0 

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

x = 1

(ix) 2x² + 5√3x + 6 = 0  

Solution:

Given quadratic equation:  2x² + 5√3x + 6 = 0      ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 2, b = 5√3, and c = 6

Now, we find the discriminant(D) = b² – 4ac

D = (5√3)² – 4(2)(6)

= 75 – 48

= 27

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

x = -√3 /2

x = -2√3 

(x) √2x² + 7x + 5√2 = 0    

Solution:

Given quadratic equation: √2x² + 7x + 5√2 = 0       ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = √2, b = 7 and c = 5√2 

Now, we find the discriminant(D) = b² – 4ac

D = (7)² – 4(√2)(5√2)

= 49 – 40

= 9 

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

x = -√2 

x = -5/√2 

(xi) 2x² – 2√2x + 1 = 0   

Solution:

Given quadratic equation: 2x² – 2√2x + 1 = 0          ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 2, b = -2√2 and c = 1

Now, we find the discriminant(D) = b² – 4ac

D = (-2√2)² – 4(2)(1)

= 8 – 8

= 0

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

x = 1/√2

(xii) 3x² – 5x + 2 = 0

Solution:

Given quadratic equation: 3x² – 5x + 2 = 0        ….(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 3, b = -5 and c = 2

Now, we find the discriminant(D) = b² – 4ac

D = (-5)² – 4(3)(2)

= 25 – 24

= 1 

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

x = 1

x = 2/3

Question 3. Solve for x:

(i) , x ≠ 2, 4

Solution:

Given: 

We can also write as

6x² – 30x + 30 = 10x² – 60x + 80

4x² – 30x + 50 = 0

2x² – 15x + 25 = 0 …(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 2, b = -15 and c = 25

Now, we find the discriminant(D) = b² – 4ac

D = (-15)² – 4(2)(25)

= 225 – 200

= 25

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

x = 5

x = 5/2

(ii) x + 1/x = 3, x ≠ 0 

Solution:

Given: x + 1/x = 3

We can also write as

x² – 3x + 1 = 0 …(1)

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 1, b = -3 and c = 1

Now, we find the discriminant(D) = b² – 4ac

D = (-3)² – 4(1)(1)

= 9 – 4

= 5 

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

(iii) , x ≠ 0, -1

Solution:

Given: 

We can also write as

(16 – x)(x + 1) = 15x

15x + 16 – x² – 15x = 0 

16 – x² = 0

x² – 16 = 0 ……(1) 

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 1, b = 0, and c = -16

Now, we find the discriminant(D) = b² – 4ac

D = (0)² – 4(1)(-16)

= 64

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

x = ±4

(iv) , x ≠ 0, 3/2, 2

Solution:

Given: 

We can also write as

(x – 2)(4x – 3) = x(2x – 3)

x² – 4x + 3 = 0 ……(1) 

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 1, b = -4, and c = 3

Now, we find the discriminant(D) = b² – 4ac

D = (-4)² – 4(1)(3)

= 4

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

x = ±3

(v) , x ≠ 3, -5

Solution:

Given: 

We can also write as

(x – 3)(x + 5) = 6 x 8

x² + 2x – 63 = 0 ……(1) 

As we know that the general form of quadratic equation is 

ax² + bx + c = 0    ….(2)

On comparing eq(1) and (2), we get  

Here, a = 1, b = 2, and c = -63

Now, we find the discriminant(D) = b² – 4ac

D = (2)² – 4(1)(-63)

= 256

As we know that for a quadratic equation having real root must satisfy the D >= 0 

Here, our equation satisfies the given condition, so it has real roots.

Now we find the real roots using the given formula:

Put the values of b, D, a in the given formula, we get

Hence, the value of x is 

x = ±9