NCERT Solutions 10th Maths Chapter 4 Quadratic Equations Exercise 4.3

NCERT Solutions For Class 10 Maths Chapter 4 Exercise 4.3 we will restart our exploration of the world of Quadratic Equations. Thus, we are providing you Chapter 4 Quadratic Equations NCERT Class 10 Maths Solutions that will help in achieving more marks. You don't have to wander and waste your precious time in finding best CBSE NCERT Solutions.

Exercise 4.3
1. Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
(i) 2x2 – 7x +3 = 0
(ii) 2x2 + x – 4 = 0
(iii) 4x2 + 4v3x + 3 = 0
(iv) 2x2 + x + 4 = 0
Answer
(i) 2x2 – 7x + 3 = 0
? 2x2 – 7x = - 3
On dividing both sides of the equation by 2, we get
? x2 – 7x/2 = -3/2
? x2 – 2 × x × 7/4 = -3/2
On adding (7/4)2 to both sides of equation, we get

? (x)2 - 2 × x × 7/4 + (7/4)2 = (7/4)2 - 3/2

? (x - 7/4)2 = 49/16 - 3/2
? (x - 7/4)2 = 25/16
? (x - 7/4) = ± 5/4
? x = 7/4 ± 5/4
? x = 7/4 + 5/4 or x = 7/4 - 5/4
? x = 12/4 or x = 2/4
? x = 3 or 1/2
(ii) 2x2 + x ñ 4 = 0
? 2x2 + x = 4
On dividing both sides of the equation, we get
? x2 + x/2 = 2
On adding (1/4)2 to both sides of the equation, we get
? (x)2 + 2 ◊ x ◊ 1/4 + (1/4)2 = 2 + (1/4)2
? (x + 1/4)2 = 33/16
? x + 1/4 = ± v33/4
? x = ± v33/4 - 1/4
? x = ± v33-1/4
? x = v33-1/4 or x = -v33-1/4
(iii) 4x2 + 4v3x + 3 = 0
? (2x)2 + 2 ◊ 2x ◊ v3 + (v3)2 = 0
? (2x + v3)2 = 0
? (2x + v3) = 0 and (2x + v3) = 0
? x = -v3/2 or x = -v3/2
(iv) 2x2 + x + 4 = 0
? 2x2 + x = -4
On dividing both sides of the equation, we get
? x2 + 1/2x = 2
? x2 + 2 ◊ x ◊ 1/4 = -2
On adding (1/4)2 to both sides of the equation, we get
? (x)2 + 2 ◊ x ◊ 1/4 + (1/4)2 = (1/4)2 - 2
? (x + 1/4)2 = 1/16 - 2
? (x + 1/4)2 = -31/16
However, the square of number cannot be negative.
Therefore, there is no real root for the given equation.

2. Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.
Answer
(i) 2x2 ñ 7x + 3 = 0
On comparing this equation with ax2 + bx + c = 0, we get
a = 2, b = -7 and c = 3
By using quadratic formula, we get
x = -b±vb2 - 4ac/2a
? x = 7±v49 - 24/4
? x = 7±v25/4
? x = 7±5/4
? x = 7+5/4 or x = 7-5/4
? x = 12/4 or 2/4
? x = 3 or 1/2
(ii) 2x2 + x - 4 = 0
On comparing this equation with ax2 + bx + c = 0, we get
a = 2, b = 1 and c = -4
By using quadratic formula, we get
x = -b±vb2 - 4ac/2a
?x = -1±v1+32/4
?x = -1±v33/4
? x = -1+v33/4 or x = -1-v33/4
(iii) 4x2 + 4v3x + 3 = 0
On comparing this equation with ax2 + bx + c = 0, we get
a = 4, b = 4v3 and c = 3
By using quadratic formula, we get
x = -b±vb2 - 4ac/2a
? x = -4v3±v48-48/8
? x = -4v3±0/8
? x = -v3/2 or x = -v3/2
(iv) 2x2 + x + 4 = 0
On comparing this equation with ax2 + bx + c = 0, we get
a = 2, b = 1 and c = 4
By using quadratic formula, we get
x = -b±vb2 - 4ac/2a
? x = -1±v1-32/4
? x = -1±v-31/4
The square of a number can never be negative.

?There is no real solution of this equation.

3. Find the roots of the following equations:
(i) x-1/x = 3, x ? 0
(ii) 1/x+4 - 1/x-7 = 11/30, x = -4, 7
Answer
(i) x-1/x = 3
? x2 - 3x -1 = 0
On comparing this equation with ax2 + bx + c = 0, we get
a = 1, b = -3 and c = -1
By using quadratic formula, we get
x = -b±vb2 - 4ac/2a
? x = 3±v9+4/2
? x = 3±v13/2
? x = 3+v13/2 or x = 3-v13/2
(ii) 1/x+4 - 1/x-7 = 11/30
? x-7-x-4/(x+4)(x-7) = 11/30
? -11/(x+4)(x-7) = 11/30
? (x+4)(x-7) = -30
? x2 - 3x - 28 = 30
? x2 - 3x + 2 = 0
? x2 - 2x - x + 2 = 0
? x(x - 2) - 1(x - 2) = 0
? (x - 2)(x - 1) = 0
? x = 1 or 2
4. The sum of the reciprocals of Rehman's ages, (in years) 3 years ago and 5 years from now is 1/3. Find his present age.
Answer
Let the present age of Rehman be x years.
Three years ago, his age was (x - 3) years.
Five years hence, his age will be (x + 5) years.
It is given that the sum of the reciprocals of Rehman's ages 3 years ago and 5 years from now is 1/3. ? 1/x-3 + 1/x-5 = 1/3
x+5+x-3/(x-3)(x+5) = 1/3
2x+2/(x-3)(x+5) = 1/3
? 3(2x + 2) = (x-3)(x+5)
? 6x + 6 = x2 + 2x - 15
? x2 - 4x - 21 = 0
? x2 - 7x + 3x - 21 = 0
? x(x - 7) + 3(x - 7) = 0
? (x - 7)(x + 3) = 0
? x = 7, -3
However, age cannot be negative.
Therefore, Rehman's present age is 7 years.
5. In a class test, the sum of Shefali's marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.
Answer
Let the marks in Maths be x.
Then, the marks in English will be 30 - x.
According to the question,
(x + 2)(30 - x - 3) = 210
(x + 2)(27 - x) = 210
? -x2 + 25x + 54 = 210
? x2 - 25x + 156 = 0
? x2 - 12x - 13x + 156 = 0
? x(x - 12) -13(x - 12) = 0
? (x - 12)(x - 13) = 0
? x = 12, 13
If the marks in Maths are 12, then marks in English will be 30 - 12 = 18
If the marks in Maths are 13, then marks in English will be 30 - 13 = 17