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

Chapter Name	RD Sharma Chapter 8 Quadratic Equations
Book Name	RD Sharma Mathematics for Class 10
Other Exercises	Exercise 8.1 Exercise 8.2 Exercise 8.3 Exercise 8.4 Exercise 8.6
Related Study	NCERT Solutions for Class 10 Maths

Exercise 8.5 Solutions

1. Write the discriminant of the following quadratic equation:

(i) 2x² - 5x + 3 = 0
(ii) x² + 2x + 4 = 0
(iii) (x - 1)(2x - 1) = 0
(iv) x² - 2x + k = 0, K ∊ R
(v) √3x² + 2√2x - 2√3 = 0
(vi) x² - x + 1 = 0
(vii) 3x² + 2x + k = 0
(viii) 4x² - 3kx + 1 = 0

Solution

(i) 2x² - 5x + 3 = 0
The given equation is in the form of ax² + bx + c = 0
here a = 2, b = -5 and c = 3
The discriminant D = b² - 4ac
⇒ (-5)² - 4×2×3
⇒ 25 - 24 = 1
∴ The discriminant of the following quadratic equation is 1.

(ii) x² + 2x + 4 = 0
The given equation is in the form of ax² + bx +c = 0
here a= 1, b = 2 and c = 4
The discriminant is D = b² - 4ac
⇒ (2)² - 4 ×1 ×4
⇒ 4 - 16 = -12
∴ The discriminant of the following quadratic equation is -12.

(iii) (x - 1)(2x - 1) = 0
The given equation is (x - 1)(2x - 1) = 0
By solving it, we get 2x² - 3x + 1 = 0
∴ This equation is in the form of ax² + bx + c = 0
here a= 2, b = -3, c = -1
The discriminant is D = b² - 4ac
⇒ (-3)² - 4 × 2 ×1
⇒ 9 - 8 = 1
∴ The discriminant D, for the following quadratic equation is 1.

(iv) x² - 2x + k = 0, K ∊ R
The given equation is in the form of ax² + bx + c = 0
Here a = 1, b = - 2, c = k [given k ∈ R]
The discriminant is D = b² - 4ac
⇒ (-2)² - 4×1 ×k
⇒ 4 - 4k
∴ The discriminant D, of the following quadratic equation is 4 - 4k, where K ∈ R

(v) √3x² + 2√2x - 2√3 = 0
The given equation is in the form of ax² + bx + c = 0
here a = √3, b = 2√2x and c = -2√3
The discriminant is D = (b² - 4ac)
⇒ (2√2)² - 4 × √3 × -2√3
⇒ 8 + 24 = 32
∴ The discriminant D, of the following quadratic equation is 32.

(vi) x² - x + 1 = 0
The given equation is in the form of ax² + bx + c = 0
here a = 1, b = - 1 and c = 1
The discriminant is D = b² - 4ac
⇒ (-1)² - 4×1×1
⇒1 - 4 = - 3
∴ The discriminant D, of the following quadratic equation is -3.

(vii) 3x² + 2x + k = 0
The given equation has 3x² + 2x + k = 0
here a = 3, b = 2, c = k
Given that the quadratic equation has real roots.
i.e., D = b² - 4ac ≥ 0
⇒ 4 - 4 ×3 ×k ≥ 0
⇒ 4 - 12k ≥ 0
⇒ 4 ≥ 12k
⇒ 4 ≤ 4/12
⇒ k ≤ 1/3
The 'k' value should not exceed 1/3 to have the real roots for the given equation.

(viii) 4x² - 3kx + 1 = 0
The given equation has 4x² - 3kx + 1 = 0
here a = 3, b = 2, c = k
Given that quadratic equation has real roots
i.e., D = b² - 4ac ≥ 0

2. In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
(i) 16x² = 24x + 1
(ii) x² + x + 2 = 0
(iii) √3x² + 10x - 8√3 = 0
(iv) 3x² - 2x + 2 = 0
(v) 2x² - 2√6x + 3 = 0
(vi) 3a² x² + 8abx + 4b² = 0 , a ≠ 0
(vii) 3x² + 2√bx - b = 0
(viii) x² - 2x + 1 = 0
(ix) 2x² + 5√3 + 6 = 0
(x) √2x² + 7x + 5√2 = 0
(xi) 2x² - 2√2x + 1 = 0
(xii) 3x² - bx + 2 = 0

Solution

(i) 16x² = 24x + 1
The given equation is in the form of 16x² - 24x + 1 = 0
Hence, the equation is in the form of ax² + bx + c = 0
Here, a = 16, b = -24, c = -1, D = b² - 4ac = (-24)² - 4 ×16 ×-1 = 576 + 64 = 640 > 0
As D > 0, the given equation has real roots, given by

(ii) x² + x + 2 = 0
The given equation is in the form of ax² + bx + c = 0
a = 1, b = 1, c = 2.
D = b² - 4ac = (1)² - 4×1×2 = 1 - 8 = -7 < 0
As Q < 0, the equation has no real roots

(iii) √3x² + 10x - 8√3 = 0
The given equation is in the form of ax² + bx + c = 0
here a = √3, b = 10 and c = -8√3

(iv) 3x² - 2x + 2 = 0
The given equation is in the form of ax² + bx + c = 0
here a = 3, b = - 2,c = 2
The discriminant Q = b² - 4ac
⇒ (-2)² - 4×3×2 = 4 - 24
⇒ - 20 < 0
Hence as Q < 0.
The given equation has no real roots.

(v) 2x² - 2√6x + 3 = 0
The given equation is in the form of ax² + bx + c = 0
here a = 2, b = -2√6, c = 3
The discriminant Q = b² - 4ac
⇒ (-2√6)² - 4×2×3 = 24 - 24
⇒ 0
As Q = 0, the given equation has real and equal roots, They are

(vi) 3a² x² +8abx + 4b² = 0, a ≠ 0
The given equation is in the form of ax² + bx +c = 0
here a = 3a² ,b = 8ab, c = 4b² [given a ≠ 0]

(vii) 3x² + 2√bx - b = 0
The given equation is in the form of ax² + bx + c = 0
here a = 3, b = 2√5, c = -5
The discriminant Q = b² - 4ac
⇒ (2√5)² -4 ×3 × -5 = 20 + 4 ×3 ×5
⇒ 20 + 60 - 80 > 0
As Q = 0, the given equation has real roots, given by

(viii) x² - 2x + 1 = 0
The given equation is in the form of ax² + bx + c = 0
Here a = 1, b = -2 and c = 1
The discriminant D = b² - 4ac
⇒ (-2)² - 4 ×1 ×1 = 0
As Q = 0, the given equation has real and equal roots

(ix) 2x² + 5√3 + 6 = 0
The given equation is in the form of ax² + bx + c = 0
here a = 2, b = 5√3, c = 6
The discriminant D = b² - 4ac
⇒ (5√3)² - 4 ×2 ×6 = 75 - 48
⇒ 27 > 0
As Q = 0 , the given equation has real roots, given by