NCERT Solutions for Class 10 Maths Chapter 4 - Quadratic Equations

Page No: 73

Exercise 4.1

1. Check whether the following are quadratic equations:
(i) (x + 1)² = 2(x - 3)
 (ii) x² - 2x = (-2)(3 - x)
 (iii) (x -2)(x - 1) = (x - 1)(x 3)
 (iv) (x - 3)(2x + 1) = x(x  5)
 (v) (2x - 1)(x - 3) = (x  5)(x - 1)
 (vi) x² + 3x + 1 = (x - 2)²
 (vii) (x + 2)³ = 2x(x² - 1)
 (viii) x³ - 4x² - x  1 = (x - 2)³

Answer

(i) (x + 1)² = 2(x - 3)
⇒ x² + 2x + 1 = 2x - 6
⇒ x² + 7 = 0
It is of the form ax² + bx + c = 0.
Hence, the given equation is quadratic equation.

(ii) x² - 2x = (-2)(3 - x)
⇒ x² - 2x = -6  2x 
⇒ x² - 4x  6 = 0
It is of the form ax²   bx   c = 0.
Hence, the given equation is quadratic equation.

(iii) (x -2)(x - 1) = (x - 1)(x  3)
⇒ x² - x - 2 = x²  2x - 3
⇒ 3x - 1 =0
It is not of the form ax²   bx   c = 0.
Hence, the given equation is not a quadratic equation.

(iv) (x - 3)(2x  1) = x(x  5)
⇒ 2x² - 5x - 3 = x²  5x
⇒  x² - 10x - 3 = 0
It is of the form ax²   bx   c = 0.
Hence, the given equation is quadratic equation.

(v) (2x - 1)(x - 3) = (x  5)(x - 1)
⇒ 2x² - 7x   3 = x²  4x - 5
⇒ x² - 11x   8 = 0
It is of the form ax²   bx   c = 0.
Hence, the given equation is quadratic equation.

(vi) x² + 3x + 1 = (x - 2)²
⇒ x² + 3x + 1 = x²  + 4 - 4x
⇒ 7x - 3 = 0
It is not of the form ax² +  bx +  c = 0.
Hence, the given equation is not a quadratic equation.

(vii) (x  +2)³ = 2x(x² - 1)
⇒ x³ + 8  +x² + 12x = 2x³ - 2x
⇒ x³ + 14x - 6x² - 8 = 0
It is not of the form ax²  + bx  + c = 0.
Hence, the given equation is not a quadratic equation.

(viii) x³ - 4x² - x + 1 = (x - 2)³
⇒  x³ - 4x² - x + 1 = x³ - 8 - 6x²  + 12x
⇒ 2x² - 13x 9 = 0
It is of the form ax²  + bx +  c = 0.
Hence, the given equation is quadratic equation.

2. Represent the following situations in the form of quadratic equations.

(i) The area of a rectangular plot is 528 m². The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

Answer

Let the breadth of the rectangular plot = x m
Hence, the length of the plot is (2x + 1) m.
Formula of area of rectangle = length × breadth = 528 m²
Putting the value of length and width, we get
(2x + 1) × x = 528
⇒ 2x² +x =528
⇒ 2x² + x - 528 = 0

(ii) The product of two consecutive positive integers is 306. We need to find the integers.

Answer

Let the first integer number = x
Next consecutive positive integer will = x + 1
Product of both integers = x × (x +1) = 306
⇒ x² + x = 306
⇒ x²  + x - 306 = 0

(iii) Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan's present age.

Answer

Let take Rohan's age = x years
Hence, his mother's age = x + 26
3 years from now
Rohan's age = x + 3
Age of Rohan's mother will = x + 26 + 3 = x + 29
The product of their ages 3 years from now will be 360 so that
(x + 3)(x + 29) = 360
⇒ x²  + 29x + 3x + 87 = 360
⇒ x²  + 32x + 87 - 360 = 0
⇒ x²  + 32x - 273 = 0

(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Answer

Let the speed of train be x km/h.
Time taken to travel 480 km = 480/x km/h

In second condition, let the speed of train = (x - 8) km/h
It is also given that the train will take 3 hours to cover the same distance.
Therefore, time taken to travel 480 km = (480/x + 3) km/h

Speed × Time = Distance

(x - 8)(480/x + 3) = 480
⇒ 480 + 3x - 3840/x - 24 = 480
⇒ 3x - 3840/x = 24
⇒ 3x² - 8x - 1280 = 0

Page No: 76

Exercise 4.2

1. Find the roots of the following quadratic equations by factorisation:
(i) x² – 3x – 10 = 0
 (ii) 2x² + x – 6 = 0
 (iii) √2 x²  7x 5√2 = 0
 (iv) 2x² – x+1/8 = 0
 (v) 100x² – 20x 1 = 0

Answer

(i) x² – 3x – 10
= x² - 5x + 2x - 10
= x(x - 5)  + 2(x - 5)
= (x - 5)(x + 2)
Roots of this equation are the values for which (x - 5)(x +  2) = 0

∴ x - 5 = 0 or x + 2 = 0

⇒ x = 5 or x = -2

(ii) 2x²  + x – 6
= 2x² + 4x - 3x - 6
= 2x(x + 2) - 3(x + 2)
= (x + 2)(2x - 3)
Roots of this equation are the values for which (x + 2)(2x - 3) = 0

∴ x + 2 = 0 or 2x - 3 = 0

⇒ x = -2 or x = 3/2

(iii) √2 x²  + 7x 6 5√2
= √2 x²  + 5x + 2x + 5√2
= x (√2x + 5)  + √2(√2x + 5)= (√2x + 5)(x + √2)
Roots of this equation are the values for which (√2x + 5)(x + √2) = 0

∴ √2x + 5 = 0 or x +√2 = 0

⇒ x = -5/√2 or x = -√2

(iv) 2x² – x + 1/8
= 1/8 (16x²  - 8x + 1)
= 1/8 (16x²  - 4x - 4x +1)
= 1/8 (4x(4x  - 1) -1(4x - 1))
= 1/8(4x - 1)²
Roots of this equation are the values for which (4x - 1)² = 0

∴ (4x - 1) = 0 or (4x - 1) = 0

⇒ x = 1/4 or x = 1/4

(v) 100x² – 20x + 1
= 100x² – 10x - 10x + 1
= 10x(10x - 1) -1(10x - 1)
= (10x - 1)²
Roots of this equation are the values for which (10x - 1)² = 0
∴ (10x - 1) = 0 or (10x - 1) = 0
⇒ x = 1/10 or x = 1/10

2. (i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. Find out how many marbles they had to start with.

Answer

Let the number of John's marbles be x.
Therefore, number of Jivanti's marble = 45 - x
After losing 5 marbles,
Number of John's marbles = x - 5
Number of Jivanti's marbles = 45 - x - 5 = 40 - x
It is given that the product of their marbles is 124.

∴ (x - 5)(40 - x) = 124

⇒ x² – 45x  324 = 0
⇒ x² – 36x - 9x + 324 = 0
⇒ x(x - 36) -9(x - 36) = 0
⇒ (x - 36)(x - 9) = 0
Either x - 36 = 0 or x - 9 = 0
⇒ x = 36 or x = 9
If the number of John's marbles = 36,
Then, number of Jivanti's marbles = 45 - 36 = 9
If number of John's marbles = 9,
Then, number of Jivanti's marbles = 45 - 9 = 36

(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs 750. Find out the number of toys produced on that day.

Answer

Let the number of toys produced be x.
∴ Cost of production of each toy = Rs (55 - x)
It is given that, total production of the toys = Rs 750

∴ x(55 - x) = 750

⇒ x² – 55x  750 = 0

⇒ x² – 25x - 30x  750 = 0

⇒ x(x - 25) -30(x - 25) = 0

⇒ (x - 25)(x - 30) = 0

Either, x -25 = 0 or x - 30 = 0
⇒ x = 25 or x = 30
Hence, the number of toys will be either 25 or 30.

3. Find two numbers whose sum is 27 and product is 182.

Answer

Let the first number be x and the second number is 27 - x.
Therefore, their product = x (27 - x)
It is given that the product of these numbers is 182.
Therefore, x(27 - x) = 182
⇒ x² – 27x  182 = 0
⇒ x² – 13x - 14x  182 = 0
⇒ x(x - 13) -14(x - 13) = 0
⇒ (x - 13)(x -14) = 0
Either x = -13 = 0 or x - 14 = 0
⇒ x = 13 or x = 14
If first number = 13, then
Other number = 27 - 13 = 14
If first number = 14, then
Other number = 27 - 14 = 13
Therefore, the numbers are 13 and 14.

4. Find two consecutive positive integers, sum of whose squares is 365.

Answer

Let the consecutive positive integers be x and x 1.
Therefore, x² + (x + 1)² = 365
⇒ x²  + x²  + 1 + 2x = 365
⇒ 2x²  + 2x - 364 = 0
⇒ x² + x - 182 = 0
⇒ x²  + 14x - 13x - 182 = 0
⇒ x(x +14) -13(x + 14) = 0
⇒ (x +14)(x - 13) = 0
Either x + 14 = 0 or x - 13 = 0,

⇒ x = - 14 or x = 13
Since the integers are positive, x can only be 13.
∴ x + 1 = 13 +1 = 14
Therefore, two consecutive positive integers will be 13 and 14.

5. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Answer

Let the base of the right triangle be x cm.
Its altitude = (x - 7) cm
From Pythagoras theorem, we have
Base²  Altitude² = Hypotenuse²
∴ x²  (x - 7)² = 132
⇒ x²  + x²  + 49 - 14x = 169
⇒ 2x² - 14x - 120 = 0
⇒ x² - 7x - 60 = 0
⇒ x² - 12x + 5x - 60 = 0
⇒ x(x - 12) + 5(x - 12) = 0
⇒ (x - 12)(x + 5) = 0
Either x - 12 = 0 or x 5 = 0,
⇒ x = 12 or x = - 5
Since sides are positive, x can only be 12.
Therefore, the base of the given triangle is 12 cm and the altitude of this triangle will be (12 - 7) cm = 5 cm.

6. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.

Answer

Let the number of articles produced be x.
Therefore, cost of production of each article = Rs (2x + 3)
It is given that the total production is Rs 90.∴ x(2x + 3) = 0
⇒ 2x²  + 3x - 90 = 0
⇒ 2x²  + 15x -12x - 90 = 0
⇒ x(2x  + 15) -6(2x + 15) = 0
⇒ (2x +15)(x - 6) = 0
Either 2x + 15 = 0 or x - 6 = 0

⇒ x = -15/2 or x = 6

As the number of articles produced can only be a positive integer, therefore, x can only be 6.
Hence, number of articles produced = 6
Cost of each article = 2 × 6  +3 = Rs 15.

Page No: 87

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  + 4√3x 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 – 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 = ± √33/4
⇒ x = ± √33/4 - 1/4
⇒ x = ± √33-1/4
⇒ x = √33-1/4 or x = -√33-1/4

(iii) 4x² + 4√3x  + 3 = 0
⇒ (2x)²  + 2 × 2x × √3 (√3)² = 0
⇒ (2x + √3)² = 0
⇒ (2x + √3) = 0 and (2x + √3) = 0
⇒ x = -√3/2 or x = -√3/2

(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 – 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±√b² - 4ac/2a
⇒ x = 7±√49 - 24/4
⇒ x = 7±√25/4
⇒ 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±√b² - 4ac/2a
⇒x = -1±√1 32/4
⇒x = -1±√33/4
∴ x = -1 + √33/4 or x = -1-√33/4

(iii) 4x² +  4√3x + 3 = 0
On comparing this equation with ax²  + bx  + c = 0, we get
a = 4, b = 4√3 and c = 3
By using quadratic formula, we get
x = -b±√b² - 4ac/2a
⇒ x = -4√3±√48-48/8
⇒ x = -4√3±0/8
∴ x = √3/2 or x = -√3/2

(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±√b² - 4ac/2a
⇒ x = -1±√1-32/4
⇒ x = -1±√-31/4
The square of a number can never be negative.
∴There is no real solution of this equation.

Page No: 88

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±√b² - 4ac/2a
⇒ x = 3±√9 4/2
⇒ x = 3±√13/2
∴ x = 3 + √13/2 or x = 3-√13/2

(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

6. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

Answer

Let the shorter side of the rectangle be x m.
Then, larger side of the rectangle = (x + 30) m

⇒ x²  (x  +30)² = (x + 60)²

⇒ x²  + x² + 900  +60x = x² + 3600 + 120x

⇒ x² - 60x - 2700 = 0

⇒ x² - 90x + 30x - 2700 = 0

⇒ x(x - 90)  +30(x -90)

⇒ (x - 90)(x + 30) = 0

⇒ x = 90, -30

However, side cannot be negative. Therefore, the length of the shorter side will be 90 m.
Hence, length of the larger side will be (90 +30) m = 120 m.

7. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

Answer

Let the larger and smaller number be x and y respectively.
According to the question,
x² - y² = 180 and y² = 8x
⇒ x² - 8x = 180
⇒ x² - 8x - 180 = 0
⇒ x² - 18x + 10x - 180 = 0
⇒ x(x - 18) + 10(x - 18) = 0
⇒ (x - 18)(x + 10) = 0
⇒ x = 18, -10

However, the larger number cannot be negative as 8 times of the larger number will be negative and hence, the square of the smaller number will be negative which is not possible.
Therefore, the larger number will be 18 only.
x = 18
∴ y² = 8x = 8 × 18 = 144
⇒ y = ±√44 = ±12
∴ Smaller number = ±12
Therefore, the numbers are 18 and 12 or 18 and - 12.

8. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Answer

Let the speed of the train be x km/hr.
Time taken to cover 360 km = 360/x hr.According to the question,
⇒ (x + 5)(360-1/x) = 360
⇒ 360 - x + 1800-5/x = 360
⇒ x²  + 5x + 10x - 1800 = 0
⇒ x(x + 45) -40(x + 45) = 0
⇒ (x  + 45)(x - 40) = 0
⇒ x = 40, -45
However, speed cannot be negative.
Therefore, the speed of train is 40 km/h.

9. Two water taps together can fill a tank in  hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Answer

Let the time taken by the smaller pipe to fill the tank be x hr.
Time taken by the larger pipe = (x - 10) hr
Part of tank filled by smaller pipe in 1 hour = 1/x

Part of tank filled by larger pipe in 1 hour = 1/x - 10

It is given that the tank can be filled in  = 75/8 hours by both the pipes together. Therefore,
1/x  1/x-10 = 8/75
x-10 x/x(x-10) = 8/75
⇒ 2x-10/x(x-10) = 8/75
⇒ 75(2x - 10) = 8x² - 80x
⇒ 150x - 750 = 8x² - 80x
⇒ 8x² - 230x + 750 = 0
⇒ 8x² - 200x - 30x  750 = 0
⇒ 8x(x - 25) -30(x - 25) = 0
⇒ (x - 25)(8x -30) = 0
⇒ x = 25, 30/8
Time taken by the smaller pipe cannot be 30/8 = 3.75 hours. As in this case, the time taken by the larger pipe will be negative, which is logically not possible.

Therefore, time taken individually by the smaller pipe and the larger pipe will be 25 and 25 - 10 =15 hours respectively.

10. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speeds of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.

Answer

Let the average speed of passenger train be x km/h.
Average speed of express train = (x +11) km/h
It is given that the time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance.

⇒ 132 × 11 = x(x + 11)

⇒ x² + 11x - 1452 = 0

⇒ x²  +  44x -33x -1452 = 0

⇒ x(x +44) -33(x + 44) = 0

⇒ (x + 44)(x - 33) = 0

⇒ x = - 44, 33

Speed cannot be negative.

Therefore, the speed of the passenger train will be 33 km/h and thus, the speed of the express train will be 33 + 11 = 44 km/h.

 11. Sum of the areas of two squares is 468 m². If the difference of their perimeters is 24 m, find the sides of the two squares.

Answer

Let the sides of the two squares be x +m and y +m. Therefore, their perimeter will be 4x and 4y respectively and their areas will be x² and y² respectively.
It is given that
4x - 4y = 24
x - y = 6
x = y +6
Also, x²  + y² = 468
⇒ (6 +  y²) +  y² = 468
⇒ 36  + y² + 12y +  y² = 468
⇒ 2y² + 12y + 432 = 0
⇒ y² + 6y - 216 = 0
⇒ y² + 18y - 12y - 216 = 0
⇒ y(y + 18) -12(y  18) = 0
⇒ (y + 18)(y - 12) = 0
⇒ y = -18, 12
However, side of a square cannot be negative.
Hence, the sides of the squares are 12 m and (12+ 6) m = 18 m.

Page No: 91

 Exercise 4.4

 1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them;
 (i) 2x² - 3x + 5 = 0
 (ii) 3x² - 4√3x + 4 = 0
 (iii) 2x² - 6x + 3 = 0

Answer

(i) Consider the equation
x² - 3x + 5 = 0
Comparing it with ax² +  bx +  c = 0, we get
a = 2, b = -3 and c = 5
Discriminant = b² - 4ac
= ( - 3)² - 4 (2) (5) = 9 - 40
= - 31
As b² - 4ac < 0,
Therefore, no real root is possible for the given equation.

(ii) 3x² - 4√3x + 4 = 0
Comparing it with ax² +  bx + c = 0, we get
a = 3, b = -4√3 and c = 4
Discriminant = b² - 4ac
= (-4√3)² - 4(3)(4)
= 48 - 48 = 0
As b² - 4ac = 0,
Therefore, real roots exist for the given equation and they are equal to each other.
And the roots will be -b/2a and -b/2a.-b/2a = -(-4√3)/2×3 = 4√3/6 = 2√3/3 = 2/√3
Therefore, the roots are 2/√3 and 2/√3.

(iii) 2x² - 6x + 3 = 0

Comparing this equation with ax² +  bx  + c = 0, we get
a = 2, b = -6, c = 3
Discriminant = b² - 4ac

= (-6)² - 4 (2) (3)
= 36 - 24 = 12
As b² - 4ac > 0,
Therefore, distinct real roots exist for this equation:

2. Find the values of k for each of the following quadratic equations, so that they have two equal roots.
 (i) 2x² + kx + 3 = 0
 (ii) kx (x - 2) + 6 = 0

Answer

(i) 2x²  + kx + 3 = 0
Comparing equation with ax² + bx + c = 0, we get
a = 2, b = k and c = 3
Discriminant = b² - 4ac

= (k)² - 4(2) (3)
= k² - 24
For equal roots,
Discriminant = 0
k² - 24 = 0
k² = 24
k = ±√24 = ±2√6

(ii) kx(x - 2) + 6 = 0
or kx² - 2kx + 6 = 0
Comparing this equation with ax² +  bx + c = 0, we get
a = k, b = - 2k and c = 6
Discriminant = b² - 4ac
= ( - 2k)2 - 4 (k) (6)
= 4k2 - 24k
For equal roots,
b2 - 4ac = 0
4k2 - 24k = 0
4k (k - 6) = 0
Either 4k = 0
or k = 6 = 0
k = 0 or k = 6
However, if k = 0, then the equation will not have the terms 'x²' and 'x'.
Therefore, if this equation has two equal roots, k should be 6 only.

3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m²? If so, find its length and breadth.

Answer
Let the breadth of mango grove be l.
Length of mango grove will be 2l.
Area of mango grove = (2l) (l)= 2l²
2l² = 800
l² = 800/2 = 400
l² - 400 =0
Comparing this equation with al² bl c = 0, we get
a = 1, b = 0, c = 400
Discriminant = b² - 4ac

= (0)² - 4 × (1) × ( - 400) = 1600
Here, b² - 4ac > 0
Therefore, the equation will have real roots. And hence, the desired rectangular mango grove can be designed.
l = ±20
However, length cannot be negative.
Therefore, breadth of mango grove = 20 m
Length of mango grove = 2 × 20 = 40 m

4. Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

Answer

Let the age of one friend be x years.
then the age of the other friend will be (20 - x) years.
4 years ago,

Age of 1st friend = (x - 4) years
Age of 2nd friend = (20 - x - 4) = (16 - x) years
A/q we get that,
(x - 4) (16 - x) = 48

16x - x² - 64 + 4x = 48
 - x²   20x - 112 = 0
x² - 20x + 112 = 0
Comparing this equation with ax²  + bx +  c = 0, we get
a = 1, b = -20 and c = 112
Discriminant = b² - 4ac
= (-20)² - 4 × 112
= 400 - 448 = -48
b² - 4ac < 0
Therefore, there will be no real solution possible for the equations. Such type of condition doesn't exist.

5. Is it possible to design a rectangular park of perimeter 80 and area 400 m2? If so find its length and breadth.

Answer

Let the length and breadth of the park be l and b.
Perimeter = 2 (l +b) = 80
l +b = 40
Or, b = 40 - l
Area = l×b = l(40 - l) = 40l - l²40l -  l² = 400
l² -  40l 400 = 0
Comparing this equation with al² +  bl +  c = 0, we get
a = 1, b = -40, c = 400
Discriminant = b² - 4ac
(-40)² - 4 × 400
= 1600 - 1600 = 0
b² - 4ac = 0
Therefore, this equation has equal real roots. And hence, this situation is possible.
Root of this equation,l = -b/2a
l = (40)/2(1) = 40/2 = 20
Therefore, length of park, l = 20 m
And breadth of park, b = 40 - l = 40 - 20 = 20 m.