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## NCERT Solutions for Class 10 Maths Quadratic Equations

**1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them.**

**(i) **

**(ii) **

**(iii) **

**Ans. (i)**

Comparing this equation with general equation ,

We get *a *= 2, *b *= −3 and *c *= 5

Discriminant = (2) (5)

= 9 – 40 = −31

Discriminant is less than 0 which means equation has no real roots.

**(ii)**

Comparing this equation with general equation ,

We get *a *= 3, *b *= and c = 4

Discriminant = = − 4 (3) (4)

= 48 – 48 = 0

Discriminant is equal to zero which means equations has equal real roots.

Applying quadratic to find roots,

Because, equation has two equal roots, it means

**(iii)**

Comparing equation with general equation ,

We get *a *= 2, *b *= −6, and *c *= 3

Discriminant = (2) (3)

= 36 – 24 = 12

Value of discriminant is greater than zero.

Therefore, equation has distinct and real roots.

Applying quadratic formula to find roots,

⇒

⇒

NCERT Solutions for Class 10 Maths Exercise 4.4

**2. Find the value of k for each of the following quadratic equations, so that they have two equal roots.**

**(i) **

**(ii) kx (x − 2) + 6 = 0**

**Ans. (i) **

We know that quadratic equation has two equal roots only when the value of discriminant is equal to zero.

Comparing equation with general quadratic equation , we get *a *= 2, *b *= *k* and *c *= 3

Discriminant = (2) (3) =

Putting discriminant equal to zero

⇒

⇒

**(ii)***kx *(*x *− 2) + 6 = 0

⇒

Comparing quadratic equation with general form , we get *a *= *k*, *b *= −2*k* and *c *= 6

Discriminant = (*k*) (6) =

We know that two roots of quadratic equation are equal only if discriminant is equal to zero.

Putting discriminant equal to zero

⇒ 4*k *(*k *− 6) = 0⇒ *k *= 0, 6

The basic definition of quadratic equation says that quadratic equation is the equation of the form , where *a *≠ 0.

Therefore, in equation , we cannot have k = 0.

Therefore, we discard k = 0.

Hence the answer is k = 6.

NCERT Solutions for Class 10 Maths Exercise 4.4

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

**Ans. **Let breadth of rectangular mango grove = *x* metres

Let length of rectangular mango grove = 2*x* metres

Area of rectangle = =

According to given condition:

⇒ = 0 = 0

Comparing equation with general form of quadratic equation , we get *a *= 1, *b *= 0 and *c *= −400

Discriminant (1) (−400) = 1600

Discriminant is greater than 0 means that equation has two disctinct real roots.

Therefore, it is possible to design a rectangular grove.

Applying quadratic formula, to solve equation,

⇒ *x *= 20, −20

We discard negative value of *x* because breadth of rectangle cannot be in negative.

Therefore, *x* = breadth of rectangle = 20 metres

Length of rectangle = 2*x *= = 40 metres

NCERT Solutions for Class 10 Maths Exercise 4.4

**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.**

**Ans. **Let age of first friend = *x* years and let age of second friend = (20 − *x*) years

Four years ago, age of first friend = (*x *− 4) years

Four years ago, age of second friend = (20 − *x*) − 4 = (16 − *x*) years

According to given condition,

(*x *− 4) (16 − *x*) = 48

⇒ = 48

⇒

⇒

Comparing equation, with general quadratic equation , we get *a *= 1, *b *= −20 and *c *= 112

Discriminant = (1) (112) = 400 – 448 = −48 < 0

Discriminant is less than zero which means we have no real roots for this equation.

Therefore, the give situation is not possible.

NCERT Solutions for Class 10 Maths Exercise 4.4

**5. Is it possible to design a rectangular park of perimeter 80 metres and area 400 ***m*^{2}. If so, find its length and breadth.

*m*

^{2}. If so, find its length and breadth.

**Ans. **Let length of park = *x* metres

We are given area of rectangular park =

Therefore, breadth of park = metres{Area of rectangle = length × *breadth*}

Perimeter of rectangular park = 2 (*length *+ *breath*) = metres

We are given perimeter of rectangle = 80 metres

According to condition:

⇒

⇒

⇒

⇒

Comparing equation, with general quadratic equation , we get *a *= 1, *b *= −40 and *c *= 400

Discriminant = (1) (400) = 1600 – 1600 = 0

Discriminant is equal to 0.

Therefore, two roots of equation are real and equal which means that it is possible to design a rectangular park of perimeter 80 metres and area .

Using quadratic formula to solve equation,

Here, both the roots are equal to 20.

Therefore, length of rectangular park = 20 metres

Breadth of rectangular park =

## NCERT Solutions for Class 10 Maths Exercise 4.4

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