**Download NCERT Solutions For Class 10 Maths Chapter 4 Exercise 4.1 – Quadratic Equations. This Exercise contains 2 questions, for which detailed answers have been provided in this note. In case you are looking at studying the remaining Exercise for Class 10 for Maths NCERT solutions for other Chapters, you can click the link at the end of this Note.**

**1. Check whether the following are quadratic equation?**

**Solution:** An equation of the form ax^{2}+bx+c = 0, a ≠ 0 and a, b, c are real numbers is called quadratic equation.

**(i) (x+1) ^{2} = 2(x-3)**

x^{2}+2x+1 = 2x-6

x^{2}+2x+1 -2x+6 = 0

x^{2}+7 = 0

Comparing above equation with ax^{2}+bx+c = 0, we have a = 1≠ 0, b = 0, c = 7

Hence the given equation is quadratic equation.

**(ii) x ^{2}-2x = (-2)(3-x)**

x^{2}-2x = -6+2x

x^{2}-2x -2x+6 = 0

x^{2}-4x+6 = 0

Comparing above equation with ax^{2}+bx+c = 0, we have a = 1≠ 0, b = -4, c = 6

Hence the given equation is quadratic equation.

**(iii) (x-2)(x+1) = (x-1)(x+3)**

x^{2}-2x+x-2 = x^{2}-x+3x-3

x^{2}-2x+x-2 -x^{2}+x-3x+3= 0

-3x+1 = 0

Comparing above equation with ax^{2}+bx+c = 0, we have a = 0, b = -3, c = 1

Hence the given equation is not quadratic equation.

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

x^{2}-6x+x-3 = x^{2}+5x

x^{2}-6x+x-3 -x^{2}-5x = 0

-10x-3 = 0

Comparing above equation with ax^{2}+bx+c = 0, we have a = 0, b = -10, c = -3

Hence the given equation is not quadratic equation.

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

2x^{2}-6x-x+3 = x^{2}+5x-x-5

2x^{2}-6x-x+3 -x^{2}-5x+x+5 = 0

x^{2}-11x+8 = 0

Comparing above equation with ax^{2}+bx+c = 0, we have a = 1 ≠ 0, b = -11, c = 8

Hence the given equation is quadratic equation.

**(vi) x ^{2}+3x+1 = (x-2)^{2}**

x^{2}+3x+1 = x^{2}-4x+4

x^{2}+3x+1 -x^{2}+4x-4= 0

7x-3 = 0

Comparing above equation with ax^{2}+bx+c = 0, we have a = 0, b = 7, c = -3

Hence the given equation is not quadratic equation.

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

x^{3}+6x^{2}+12x+8 = 2x^{3}-2x

x^{3}+6x^{2}+12x+8 -2x^{3}+2x = 0

x^{3 }– 6x^{2}-14x-8 = 0

It is not in the form of ax^{2}+bx+c = 0s

Hence the given equation is not a quadratic equation.

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

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

x^{3}-4x^{2}-x+1 -x^{3}+6x^{2}-12x+8= 0

2x^{2}-13x+9 = 0

Comparing above equation with ax^{2}+bx+c = 0, we have a = 2 ≠ 0, b = -13, c = 9

Hence the given equation is quadratic equation.

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

**(i) The area of rectangular plot is 528 m ^{2}. The length of the plot (in meters) is one more than twice its breadth. We need to find the length and breadth of the plot .**

**Solution:** Let the breadth of the plot is x meters

Then, the length is (2x + 1) meters.

Area = length × breadth

528 = x × (2x + 1)

528 = 2x^{2} + x

2x^{2} + x – 528 = 0

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

**Solution: **Let x and x+1 be two consecutive integers.

Then, x(x + 1) = 306

x^{2 }+ x – 306 = 0

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

**Solution:** Let x be the Rohan’s present age. Then, his mother’s present age is x+26

After 3 years theirs ages will be x+3 and x+26+3 = x+29 respectively.

Hence,

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

x^{2}+3x+29x+87 = 360

x^{2}+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.**

**Solution:** Let the speed of the train be x km/h

Then it covers 480 km in hours.

If the speed is x-8 km/hr then it takes hours.

Hence, – = 3

= 3

480×8 = 3x(x-8)

x^{2}-8x = 160×8

x^{2}– 8x -1280 = 0

**NCERT Solutions for Class 10 Maths Chapter 4 Exercise 4.1 – Quadratic Equations, has been designed by the NCERT to test the knowledge of the student on the topic – Quadratic Equations**

**Download NCERT Solutions For Class 10 Maths Chapter 4 Exercise 4.1 – Quadratic Equations**