# NCERT Solutions Quadratic Equations Class 10

Class 10Quadratic Equations Class 10- Quadratic equations are an essential part of mathematics and play a significant role in various fields of science and engineering. Studying quadratic equations in Class 10 is crucial as it helps in understanding fundamental concepts in algebra, such as polynomials and their properties. Moreover, quadratic equations offer a foundation for solving real-life problems, including calculating distances and determining maximum and minimum values of a function. By studying quadratic equations, students develop their analytical and problem-solving skills, which are necessary for further studies and professional careers in science, technology, engineering, and mathematics (STEM).

## Answers to NCERT Class 10 Maths Chapter 4 – Quadratic Equations Class 10

### Exercise 4.1 Page: 73

**1. Check whether the following are quadratic equations:**

**(i) (x + 1) ^{2} = 2(x – 3)**

**(ii) x ^{2} – 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) x ^{2} + 3x + 1 = (x – 2)^{2}**

**(vii) (x + 2) ^{3} = 2x (x^{2} – 1)**

**(viii) x ^{3} – 4x^{2} – x + 1 = (x – 2)^{3}**

**Solutions:**

(i) Given,

(x + 1)^{2} = 2(x – 3)

By using the formula for (a+b)^{2 }= a^{2}+2ab+b^{2}

⇒ x^{2} + 2x + 1 = 2x – 6

⇒ x^{2} + 7 = 0

The above equation is in the form of ax^{2} + bx + c = 0

Therefore, the given equation is a quadratic equation.

(ii) Given, x^{2} – 2x = (–2) (3 – x)

⇒ x^{2 }–^{ }2x = -6 + 2x

⇒ x^{2 }– 4x + 6 = 0

The above equation is in the form of ax^{2} + bx + c = 0

Therefore, the given equation is a quadratic equation.

(iii) Given, (x – 2)(x + 1) = (x – 1)(x + 3)

By multiplication,

⇒ x^{2 }– x – 2 = x^{2 }+ 2x – 3

⇒ 3x – 1 = 0

The above equation is not in the form of ax^{2} + bx + c = 0

Therefore, the given equation is not a quadratic equation.

(iv) Given, (x – 3)(2x +1) = x(x + 5)

By multiplication,

⇒ 2x^{2 }– 5x – 3 = x^{2 }+ 5x

⇒ x^{2 }– 10x – 3 = 0

The above equation is in the form of ax^{2} + bx + c = 0

Therefore, the given equation is a quadratic equation.

(v) Given, (2x – 1)(x – 3) = (x + 5)(x – 1)

By multiplication,

⇒ 2x^{2 }– 7x + 3 = x^{2 }+ 4x – 5

⇒ x^{2 }– 11x + 8 = 0

The above equation is in the form of ax^{2} + bx + c = 0.

Therefore, the given equation is a quadratic equation.

(vi) Given, x^{2} + 3x + 1 = (x – 2)^{2}

By using the formula for (a-b)^{2}=a^{2}-2ab+b^{2}

⇒ x^{2} + 3x + 1 = x^{2} + 4 – 4x

⇒ 7x – 3 = 0

The above equation is not in the form of ax^{2} + bx + c = 0

Therefore, the given equation is not a quadratic equation.

(vii) Given, (x + 2)^{3} = 2x(x^{2} – 1)

By using the formula for (a+b)^{3 }= a^{3}+b^{3}+3ab(a+b)

⇒ x^{3} + 8 + x^{2} + 12x = 2x^{3} – 2x

⇒ x^{3} + 14x – 6x^{2} – 8 = 0

The above equation is not in the form of ax^{2} + bx + c = 0

Therefore, the given equation is not a quadratic equation.

(viii) Given, x^{3} – 4x^{2} – x + 1 = (x – 2)^{3}

By using the formula for (a-b)^{3 }= a^{3}-b^{3}-3ab(a-b)

⇒ x^{3} – 4x^{2} – x + 1 = x^{3} – 8 – 6x^{2 } + 12x

⇒ 2x^{2} – 13x + 9 = 0

The above equation is in the form of ax^{2} + bx + c = 0

Therefore, the given equation is a quadratic equation.

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

**(i) The area of a rectangular plot is 528 m ^{2}. 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.**

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

**(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.**

**(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**

**Solutions:**

(i) Let us consider,

The breadth of the rectangular plot = x m

Thus, the length of the plot = (2x + 1) m

As we know,

Area of rectangle = length × breadth = 528 m^{2}

Putting the value of the length and breadth of the plot in the formula, we get,

(2x + 1) × x = 528

⇒ 2x^{2} + x =528

⇒ 2x^{2} + x – 528 = 0

Therefore, the length and breadth of the plot satisfy the quadratic equation, 2x^{2} + x – 528 = 0, which is the required representation of the problem mathematically.

(ii) Let us consider,

The first integer number = x

Thus, the next consecutive positive integer will be = x + 1

Product of two consecutive integers = x × (x +1) = 306

⇒ x^{2 }+ x = 306

⇒ x^{2 }+ x – 306 = 0

Therefore, the two integers x and x+1 satisfy the quadratic equation, x^{2 }+ x – 306 = 0, which is the required representation of the problem mathematically.

(iii) Let us consider,

Age of Rohan’s = x years

Therefore, as per the given question,

Rohan’s mother’s age = x + 26

After 3 years,

Age of Rohan’s = x + 3

Age of Rohan’s mother will be = x + 26 + 3 = x + 29

The product of their ages after 3 years will be equal to 360, such that

(x + 3)(x + 29) = 360

⇒ x^{2} + 29x + 3x + 87 = 360

⇒ x^{2} + 32x + 87 – 360 = 0

⇒ x^{2} + 32x – 273 = 0

Therefore, the age of Rohan and his mother satisfies the quadratic equation, x^{2} + 32x – 273 = 0, which is the required representation of the problem mathematically.

(iv) Let us consider,

The speed of the train = *x* km/h

And

Time taken to travel 480 km = 480/x km/hr

As per second condition, the speed of train = (*x* – 8) km/h

Also given, the train will take 3 hours to cover the same distance.

Therefore, time taken to travel 480 km = (480/x)+3 km/h

As we know,

Speed × Time = Distance

Therefore,

(*x* – 8)(480/*x* )+ 3 = 480

⇒ 480 + 3*x* – 3840/*x* – 24 = 480

⇒ 3*x* – 3840/*x* = 24

⇒ *x*^{2 }– 8*x* – 1280 = 0

Therefore, the speed of the train satisfies the quadratic equation, *x*^{2 }– 8*x* – 1280 = 0, which is the required representation of the problem mathematically.

### Exercise 4.2 Page: 76

**1. Find the roots of the following quadratic equations by factorisation:**

**(i) x ^{2} – 3x – 10 = 0(ii) 2x^{2} + x – 6 = 0(iii) √2 x^{2} + 7x + 5√2 = 0(iv) 2x^{2} – x +1/8 = 0(v) 100x^{2} – 20x + 1 = 0**

**Solutions:**

(i) Given, *x*^{2} – 3*x* – 10 =0

Taking L.H.S.,

=>*x*^{2} – 5*x* + 2*x* – 10

=>*x*(*x *– 5) + 2(*x* – 5)

=>(*x* – 5)(*x* + 2)

The roots of this equation,* x*^{2} – 3*x* – 10 = 0 are the values of x for which (*x* – 5)(*x* + 2) = 0

Therefore, *x* – 5 = 0 or *x* + 2 = 0

=> *x* = 5 or *x* = -2

(ii) Given, 2*x*^{2} + *x* – 6 = 0

Taking L.H.S.,

=> 2*x*^{2} + 4*x* – 3*x* – 6

=> 2*x*(*x* + 2) – 3(*x* + 2)

=> (*x* + 2)(2*x* – 3)

The roots of this equation, 2*x*^{2} + *x* – 6=0 are the values of x for which (*x* *x* + 2)(2*x* – 3) = 0

Therefore, *x* + 2 = 0 or 2*x* – 3 = 0

=> *x* = -2 or *x* = 3/2

(iii) √2 *x*^{2} + 7*x* + 5√2=0

Taking L.H.S.,

=> √2 *x*^{2 }+ 5*x* + 2*x* + 5√2

=> *x* (√2*x* + 5) + √2(√2*x* + 5)= (√2*x* + 5)(*x *+ √2)

The roots of this equation, √2 *x*^{2} + 7*x* + 5√2=0 are the values of x for which (√2*x* + 5)(*x *+ √2) = 0

Therefore, √2*x* + 5 = 0 or *x* + √2 = 0

=> *x* = -5/√2 or *x* = -√2

(iv) 2*x*^{2} – *x* +1/8 = 0

Taking L.H.S.,

=1/8 (16*x*^{2 }– 8*x* + 1)

= 1/8 (16*x*^{2 }– 4*x* -4*x* + 1)

= 1/8 (4*x*(4*x*^{ }– 1) -1(4*x* – 1))

= 1/8 (4*x *– 1)^{2}

The roots of this equation, 2*x*^{2} – *x* + 1/8 = 0, are the values of x for which (4*x *– 1)^{2}= 0

Therefore, (4*x* – 1) = 0 or (4*x* – 1) = 0

⇒ *x* = 1/4 or *x* = 1/4

(v) Given, 100x^{2} – 20x + 1=0

Taking L.H.S.,

= 100x^{2} – 10x – 10x + 1

= 10x(10x – 1) -1(10x – 1)

= (10x – 1)^{2}

The roots of this equation, 100x^{2} – 20x + 1=0, are the values of x for which (10x – 1)^{2}= 0

∴ (10x – 1) = 0 or (10x – 1) = 0

⇒x = 1/10 or x = 1/10

**2. Solve the problems given in Example 1.**

**Represent the following situations mathematically:**

**(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. We would like to find out how many marbles they had to start with.**

**(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. We would like to find out the number of toys produced on that day.**

**Solutions:**

(i) Let us say the number of marbles John has = *x*

Therefore, the number of marble Jivanti has = 45 – *x*

After losing 5 marbles each,

Number of marbles John has = *x* – 5

Number of marble Jivanti has = 45 – *x* – 5 = 40 –* x*

Given that the product of their marbles is 124.

∴ (*x *– 5)(40 – *x*) = 124

⇒ *x*^{2} – 45*x* + 324 = 0

⇒ *x*^{2} – 36*x* – 9*x* + 324 = 0

⇒ *x*(*x* – 36) -9(*x* – 36) = 0

⇒ (*x* – 36)(*x* – 9) = 0

Thus, we can say,

*x* – 36 = 0 or *x* – 9 = 0

⇒ *x* = 36 or *x* = 9

*Therefore,*

If John’s marbles = 36

Then, Jivanti’s marbles = 45 – 36 = 9

And if John’s marbles = 9

Then, Jivanti’s marbles = 45 – 9 = 36

(ii) Let us say the number of toys produced in a day is *x*.

Therefore, cost of production of each toy = Rs(55 – *x*)

Given the total cost of production of the toys = Rs 750

∴ *x*(55 – *x*) = 750

⇒ *x*^{2} – 55*x* + 750 = 0

⇒ *x*^{2} – 25*x* – 30*x* + 750 = 0

⇒* x*(*x* – 25) -30(*x* – 25) = 0

⇒ (*x* – 25)(*x* – 30) = 0

Thus, either *x* -25 = 0 or *x* – 30 = 0

⇒ *x* = 25 or *x* = 30

Hence, the number of toys produced in a day will be either 25 or 30.

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

**Solution:**

Let us say the first number is x, and the second number is 27 – x.

Therefore, the product of two numbers

x(27 – x) = 182

⇒ x^{2} – 27x – 182 = 0

⇒ x^{2} – 13x – 14x + 182 = 0

⇒ x(x – 13) -14(x – 13) = 0

⇒ (x – 13)(x -14) = 0

Thus, either, x = -13 = 0 or x – 14 = 0

⇒ x = 13 or x = 14

Therefore, if first number = 13, then second number = 27 – 13 = 14

And if first number = 14, then second number = 27 – 14 = 13

Hence, the numbers are 13 and 14.

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

**Solution:**

Let us say the two consecutive positive integers are *x* and *x* + 1.

Therefore, as per the given questions,

*x*^{2} + (*x* + 1)^{2} = 365

⇒ *x*^{2 }+ *x*^{2 }+ 1 + 2*x* = 365

⇒ 2*x*^{2} + 2x – 364 = 0

⇒ *x*^{2 }+ *x *– 182 = 0

⇒ *x*^{2 }+ 14*x* – 13*x* – 182 = 0

⇒ *x*(*x* + 14) -13(*x* + 14) = 0

⇒ (*x* + 14)(*x* – 13) = 0

Thus, either, *x* + 14 = 0 or *x* – 13 = 0,

⇒ *x* = – 14 or *x* = 13

Since the integers are positive, *x* can be 13 only.

∴ *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.**

**Solution:**

Let us say the base of the right triangle is x cm.

Given, the altitude of right triangle = (x – 7) cm

From Pythagoras’ theorem, we know,

Base^{2} + Altitude^{2} = Hypotenuse^{2}

∴ x^{2 }+ (x – 7)^{2} = 13^{2}

⇒ x^{2 }+ x^{2 }+ 49 – 14x = 169

⇒ 2x^{2 }– 14x – 120 = 0

⇒ x^{2 }– 7x – 60 = 0

⇒ x^{2 }– 12x + 5x – 60 = 0

⇒ x(x – 12) + 5(x – 12) = 0

⇒ (x – 12)(x + 5) = 0

Thus, either x – 12 = 0 or x + 5 = 0,

⇒ x = 12 or x = – 5

Since sides cannot be negative, 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.**

**Solution:**

Let us say the number of articles produced is *x*.

Therefore, cost of production of each article = Rs (2*x* + 3)

Given the total cost of production is Rs.90

∴ *x*(2*x* + 3) = 90

⇒ 2*x*^{2 }+ 3*x* – 90 = 0

⇒ 2*x*^{2 }+ 15*x* -12*x* – 90 = 0

⇒ *x*(2*x* + 15) -6(2*x* + 15) = 0

⇒ (2*x* + 15)(*x* – 6) = 0

Thus, either 2*x* + 15 = 0 or *x* – 6 = 0

⇒ *x* = -15/2 or *x* = 6

As the number of articles produced can only be a positive integer, x can only be 6.

Hence, the number of articles produced = 6

Cost of each article = 2 × 6 + 3 = Rs 15

### Exercise 4.3 Page: 87

**1. Find the roots of the following quadratic equations, if they exist, by the method of completing the square:**

**(i) 2 x^{2} – 7x +3 = 0**

**(ii) 2 x^{2} + x – 4 = 0**

(iii) 4x^{2} + 4√3x + 3 = 0

**(iv) 2 x^{2} + x + 4 = 0**

**Solutions:**

**(i) 2 x^{2} – 7x + 3 = 0**

⇒ 2*x*^{2} – 7*x *= – 3

Dividing by 2 on both sides, we get

⇒ x^{2} -7x/2 = -3/2

⇒ x^{2 }-2 × x ×7/4 = -3/2

On adding (7/4)^{2} to both sides of the 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)^{2} = ±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 x = 1/2

**(ii) 2x ^{2} + x – 4 = 0**

⇒ 2x^{2} + x = 4

Dividing both sides of the equation by 2, we get

⇒ x^{2} +x/2 = 2

Now 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

Therefore, either x = (√33-1)/4 or x = (-√33-1)/4

**(iii) 4x ^{2} + 4√3x + 3 = 0**

Converting the equation into a^{2}+2ab+b^{2 }form, we get,

⇒ (2x)^{2} + 2 × 2x × √3 + (√3)^{2} = 0

⇒ (2x + √3)^{2} = 0

⇒ (2x + √3) = 0 and (2x + √3) = 0

Therefore, either x = -√3/2 or x = -√3/2

**(iv) 2x ^{2} + x + 4 = 0**

⇒ 2x^{2} + x = -4

Dividing both sides of the equation by 2, we get

⇒ x^{2} + 1/2x = 2

⇒ x^{2} + 2 × x × 1/4 = -2

By 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

As we know, the square of numbers cannot be negative.

Therefore, there is no real root for the given equation, 2x^{2} + x + 4 = 0

**2. Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.**

**(i) 2x ^{2} – 7x + 3 = 0**

On comparing the given equation with ax^{2} + bx + c = 0, we get,

a = 2, b = -7 and c = 3

By using the quadratic formula, we get,

⇒ 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) 2x ^{2} + x – 4 = 0**

On comparing the given equation with ax^{2} + bx + c = 0, we get,

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

By using the quadratic formula, we get,

⇒x = (-1±√1+32)/4

⇒x = (-1±√33)/4

∴ x = (-1+√33)/4 or x = (-1-√33)/4

**(iii) 4x ^{2} + 4√3x + 3 = 0**

On comparing the given equation with ax^{2} + bx + c = 0, we get

a = 4, b = 4√3 and c = 3

By using the quadratic formula, we get,

⇒ x = (-4√3±√48-48)/8

⇒ x = (-4√3±0)/8

∴ x = -√3/2 or x = -√3/2

**(iv) 2x ^{2} + x + 4 = 0**

On comparing the given equation with ax^{2} + bx + c = 0, we get,

a = 2, b = 1 and c = 4

By using the quadratic formula, we get

⇒ x = (-1±√1-32)/4

⇒ x = (-1±√-31)/4

As we know, the square of a number can never be negative. Therefore, there is no real solution for the given 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**

**Solution:**

(i) x-1/x = 3

⇒ x^{2} – 3x -1 = 0

On comparing the given equation with ax^{2} + bx + c = 0, we get

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

By using the quadratic formula, we get,

⇒ 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

⇒ x^{2} – 3x – 28 = 30

⇒ x^{2} – 3x + 2 = 0

We can solve this equation by factorisation method now.

⇒ x^{2} – 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 age (in years) 3 years ago and 5 years from now is 1/3. Find his present age.**

**Solution:**

Let us say the present age of Rahman is *x* years.

Three years ago, Rehman’s age was (*x* – 3) years.

Five years after, his age will be (*x* + 5) years.

Given the sum of the reciprocals of Rehman’s ages 3 years ago and after 5 years is equal to 1/3.

∴ 1/*x*-3 + 1/*x*-5 = 1/3

*(x*+5+*x*-3)/(*x*-3)(*x*+5) = 1/3

(2*x*+2)/(*x*-3)(*x*+5) = 1/3

⇒ 3(2*x* + 2) = (*x*-3)(*x*+5)

⇒ 6*x* + 6 = *x*^{2} + 2*x* – 15

⇒ *x*^{2} – 4*x* – 21 = 0

⇒ *x*^{2} – 7*x* + 3*x* – 21 = 0

⇒ *x*(*x* – 7) + 3(*x* – 7) = 0

⇒ (*x* – 7)(*x* + 3) = 0

⇒ *x* = 7, -3

As we know, age cannot be negative.

Therefore, Rahman’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.**

**Solution:**

Let us say the marks of Shefali in Maths be x.

Then, the marks in English will be 30 – x.

As per the given question,

(x + 2)(30 – x – 3) = 210

(x + 2)(27 – x) = 210

⇒ -x^{2} + 25x + 54 = 210

⇒ x^{2} – 25x + 156 = 0

⇒ x^{2} – 12x – 13x + 156 = 0

⇒ x(x – 12) -13(x – 12) = 0

⇒ (x – 12)(x – 13) = 0

⇒ x = 12, 13

Therefore, if the marks in Maths are 12, then marks in English will be 30 – 12 = 18, and 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.**

**Solution:**

Let us say the shorter side of the rectangle is *x* m.

Then, larger side of the rectangle = (*x* + 30) m

As given, the length of the diagonal is = x + 30 m

Therefore,

⇒ *x*^{2} + (*x* + 30)^{2} = (*x* + 60)^{2}

⇒ *x*^{2} + *x*^{2} + 900 + 60*x* = *x*^{2} + 3600 + 120*x*

⇒ *x*^{2} – 60*x* – 2700 = 0

⇒ *x*^{2} – 90*x* + 30*x* – 2700 = 0

⇒ *x*(*x *– 90) + 30(*x* -90) = 0

⇒ (*x *– 90)(*x* + 30) = 0

⇒ *x *= 90, -30

However, the side of the field cannot be negative. Therefore, the length of the shorter side will be 90 m.

And the length of the larger side will be (90 + 30) m = 120 m.

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

**Solution:**

Let us say the larger and smaller number be *x* and *y,* respectively.

As per the question given,

*x*^{2 }– *y*^{2} = 180 and *y*^{2} = 8*x*

⇒ *x*^{2 }– 8*x* = 180

⇒ *x*^{2 }– 8*x* – 180 = 0

⇒ *x*^{2 }– 18*x* + 10*x* – 180 = 0

⇒ *x*(*x* – 18) +10(*x* – 18) = 0

⇒ (*x* – 18)(*x* + 10) = 0

⇒ *x* = 18, -10

However, the larger number cannot be considered a negative number, as 8 times 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

∴ *y*^{2} = 8x = 8 × 18 = 144

⇒ *y* = ±√144 = ±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.**

**Solution:**

It is given that

Distance = 360 km

Consider x as the speed, then the time taken

t = 360/x

If the speed is increased by 5 km/h, the speed will be (x + 5) km/h.

Distance will be the same.

t = 360/(x + 5)

We know that

Time with original speed – Time with increased speed = 1

360/x – 360/(x + 5) = 1

LCM = x (x + 5)

[360 (x + 5) – 360x]/x(x + 5) = 1

360 x + 1800 – 360x = x (x + 5)

x^{2} + 5x = 1800

x^{2} + 5x – 1800 = 0

x^{2} + 45x – 40x – 1800 = 0

x (x + 45) – 40 (x + 45) = 0

(x – 40) (x + 45) = 0

x = 40 km/hr

As we know, the value of speed cannot be negative.

Therefore, the speed of the train is 40 km/h.

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

**Solution:**

Let the time taken by the smaller pipe to fill the tank = x hr.

Time taken by the larger pipe = (*x* – 10) hr

Part of the tank filled by smaller pipe in 1 hour = 1/*x*

Part of the tank filled by larger pipe in 1 hour = 1/(*x *– 10)

As given, 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

⇒ 2*x*-10/*x*(*x*-10) = 8/75

⇒ 75(2*x* – 10) = 8*x*^{2} – 80*x*

⇒ 150*x* – 750 = 8*x*^{2} – 80*x*

⇒ 8*x*^{2} – 230*x* +750 = 0

⇒ 8*x*^{2} – 200*x* – 30*x* + 750 = 0

⇒ 8*x*(*x* – 25) -30(*x* – 25) = 0

⇒ (*x* – 25)(8*x* -30) = 0

⇒ *x* = 25, 30/8

Time taken by the smaller pipe cannot be 30/8 = 3.75 hours, as the time taken by the larger pipe will become negative, which is logically not possible.

Therefore, the 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 speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.**

**Solution:**

Let us say the average speed of the passenger train = *x* km/h.

Average speed of express train = (*x* + 11) km/h

Given the time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance. Therefore,

(132/x) – (132/(x+11)) = 1

132(x+11-x)/(x(x+11)) = 1

132 × 11 /(x(x+11)) = 1

⇒ 132 × 11 = *x*(*x* + 11)

⇒ *x*^{2} + 11*x* – 1452 = 0

⇒ *x*^{2} + 44*x* -33*x* -1452 = 0

⇒ *x*(*x* + 44) -33(*x* + 44) = 0

⇒ (*x* + 44)(*x* – 33) = 0

⇒ *x* = – 44, 33

As we know, 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 m ^{2}. If the difference between their perimeters is 24 m, find the sides of the two squares.**

**Solution:**

Let the sides of the two squares be *x* m and *y* m.

Therefore, their perimeter will be 4*x* and 4*y,* respectively

And the area of the squares will be *x*^{2} and *y*^{2,} respectively.

Given,

4*x* – 4*y* = 24

*x* – *y* = 6

*x* = *y* + 6

Also, *x*^{2 }+^{ }*y*^{2} = 468

⇒ (6 + *y*^{2}) + *y*^{2} = 468

⇒ 36 + *y*^{2} + 12*y* + *y*^{2} = 468

⇒ 2*y*^{2} + 12*y* + 432 = 0

⇒ *y*^{2} + 6y – 216 = 0

⇒ *y*^{2} + 18*y* – 12*y* – 216 = 0

⇒ *y*(*y *+18) -12(*y* + 18) = 0

⇒ (*y* + 18)(*y* – 12) = 0

⇒ *y* = -18, 12

As we know, the side of a square cannot be negative.

Hence, the sides of the squares are 12 m and (12 + 6) m = 18 m.

### Exercise 4.4 Page: 91 -Quadratic Equations Class 10

**1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them.(i) 2 x^{2} – 3x + 5 = 0(ii) 3x^{2} – 4√3x + 4 = 0(iii) 2x^{2} – 6x + 3 = 0**

**Solutions:**

(i) Given,

*2x*^{2} – 3*x* + 5 = 0

Comparing the equation with *ax*^{2} + *bx *+ *c* = 0, we get

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

We know, Discriminant = *b*^{2} – 4*ac*

*= *( – 3)^{2} – 4 (2) (5) = 9 – 40

= – 31

As you can see, b^{2} – 4ac < 0

Therefore, no real root is possible for the given equation, *2x*^{2} – 3*x* + 5 = 0

(ii) 3*x*^{2} – 4√3*x* + 4 = 0

Comparing the equation with *ax*^{2} + *bx *+ *c* = 0, we get

*a* = 3, *b* = -4√3 and *c* = 4

We know, Discriminant = *b*^{2} – 4*ac*

= (-4√3)^{2 }– 4(3)(4)

= 48 – 48 = 0

As *b*^{2} – 4*ac* = 0,

Real roots exist for the given equation, and they are equal to each other.

Hence, the roots will be –*b*/2*a* and –*b*/2*a*.

–*b*/2*a *= -(-4√3)/2×3 = 4√3/6 = 2√3/3 = 2/√3

Therefore, the roots are 2/√3 and 2/√3.

(iii) 2*x*^{2} – 6*x* + 3 = 0

Comparing the equation with *ax*^{2} + *bx *+ *c* = 0, we get

*a* = 2, *b* = -6, *c* = 3

As we know, Discriminant = *b*^{2} – 4*ac*

= (-6)^{2} – 4 (2) (3)

= 36 – 24 = 12

As *b*^{2} – 4*ac* > 0,

Therefore, there are distinct real roots that exist for this equation, 2*x*^{2} – 6*x* + 3 = 0

= (-(-6) ± √(-6^{2}-4(2)(3)) )/ 2(2)

= (6±2√3 )/4

= (3±√3)/2

Therefore, the roots for the given equation are (3+√3)/2 and (3-√3)/2

**2. Find the values of k for each of the following quadratic equations so that they have two equal roots.(i) 2x^{2} + kx + 3 = 0(ii) kx (x – 2) + 6 = 0**

**Solutions:**

(i) 2*x*^{2} + *kx* + 3 = 0

Comparing the given equation with *ax*^{2} + *bx *+ *c* = 0, we get,

*a* = 2, *b* = k and *c* = 3

As we know, Discriminant = *b*^{2} – 4*ac*

= (*k*)^{2} – 4(2) (3)

= *k*^{2} – 24

For equal roots, we know,

Discriminant = 0

*k*^{2} – 24 = 0

*k*^{2} = 24

k = ±√24 = ±2√6

(ii) *kx*(*x* – 2) + 6 = 0

or *kx*^{2} – 2*kx* + 6 = 0

Comparing the given equation with *ax*^{2} + *bx *+ *c* = 0, we get

*a* = *k*, *b* = – 2*k* and *c* = 6

We know, Discriminant = *b*^{2} – 4*ac*

= ( – 2*k*)^{2} – 4 (*k*) (6)

= 4*k ^{2}* – 24

*k*

For equal roots, we know,

*b*^{2} – 4*ac* = 0

4*k*^{2} – 24*k* = 0

4*k* (*k* – 6) = 0

Either 4*k* = 0 or *k* = 6 = 0

*k* = 0 or *k* = 6

However, if *k* = 0, then the equation will not have the terms ‘*x*^{2}‘ 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 m ^{2}? If so, find its length and breadth.**

**Solution:**

Let the breadth of the mango grove be *l*.

The length of the mango grove will be 2*l*.

Area of the mango grove = (2*l*) (*l*)= 2*l*^{2}

2*l*^{2 }= 800

*l*^{2 }= 800/2 = 400

*l*^{2 }– 400 =0

Comparing the given equation with *ax*^{2} + *bx *+ *c* = 0, we get

*a* = 1, *b* = 0, *c* = 400

As we know, Discriminant = *b*^{2} – 4*ac*

=> (0)^{2} – 4 × (1) × ( – 400) = 1600

Here, *b*^{2} – 4*ac* > 0

Thus, the equation will have real roots. And hence, the desired rectangular mango grove can be designed.

*l *= ±20

As we know, the value of length cannot be negative.

Therefore, the breadth of the 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 age in years was 48.**

**Solution:**

Let’s say the age of one friend is x years.

Then, the age of the other friend will be (20 – x) years.

Four years ago,

Age of First friend = (*x* – 4) years

Age of Second friend = (20 – *x* – 4) = (16 –* x*) years

As per the given question, we can write,

(*x* – 4) (16 – *x*) = 48

16*x – x*^{2} – 64 + 4*x* = 48

* – x*^{2} + 20*x – *112 = 0

*x*^{2} – 20*x + *112 = 0

Comparing the equation with *ax*^{2} + *bx *+ *c* = 0, we get

*a* = *1*, *b* = -2*0* and *c* = 112

Discriminant = *b*^{2} – 4*ac*

*= (-*20*)*^{2} – 4 × 112

= 400 – 448 = -48

*b*^{2} – 4*ac *< 0

Therefore, there will be no real solution possible for the equations. Hence, the condition doesn’t exist.

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

**Solution:**

Let the length and breadth of the park be *l *and b.

Perimeter of the rectangular park = 2 (*l + b*) = 80

*So, l + b *= 40

Or, *b* = 40 – *l*

Area of the rectangular park = *l×b = l(40 – l) = *40*l *– *l*^{2 }= 400

*l*^{2}* *–^{ }40*l* + 400* *= 0, which is a quadratic equation.

Comparing the equation with *ax*^{2} + *bx *+ *c* = 0, we get

*a* = 1, *b* = -40, *c* = 400

Since, Discriminant = *b*^{2} – 4*ac*

*=(-*40*)*^{2} – 4 × 400

= 1600 – 1600 = 0

Thus,* b*^{2} – 4*ac *= 0

Therefore, this equation has equal real roots. Hence, the situation is possible.

The root of the equation,

*l* = –*b*/2*a*

*l* = -(-40)/2(1) = 40/2 = 20

Therefore, the length of the rectangular park, *l *= 20 m

And the breadth of the park, *b *= 40 – *l *= 40 – 20 = 20 m.

### Frequently Asked Questions

### What is a quadratic equation?

A: A quadratic equation is a second-degree polynomial equation in one variable, which can be written in the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.

### What is the formula to solve a quadratic equation?

A : The quadratic formula is used to solve a quadratic equation and is given by x = (-b ± √(b^2 - 4ac)) / 2a.

### How can I determine the nature of the roots of a quadratic equation?

A: The nature of the roots of a quadratic equation can be determined by calculating the discriminant, which is given by b^2 - 4ac. If the discriminant is greater than 0, the equation has two distinct real roots. If it is equal to 0, the equation has one real root (a repeated root). If it is less than 0, the equation has two complex conjugate roots.

### How can I solve a quadratic equation by factorization method?

A: To solve a quadratic equation by factorization method, we need to factorize the quadratic expression into two linear factors and then equate each factor to zero to find the roots.

### How can I solve a quadratic equation using completing the square method?

A: To solve a quadratic equation using completing the square method, we need to add and subtract a suitable constant to the quadratic expression to make it a perfect square trinomial. Then we can take the square root of both sides to solve for the variable.

### What is the significance of the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation determines the nature of the roots of the equation. It tells us whether the roots are real or complex, and whether they are distinct or repeated.

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