Class 10 Mathematics

Chapter 4 Solutions — Quadratic Equations

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Solutions

Overview

Step-by-step NCERT solutions for Quadratic Equations (Chapter 4, CBSE Class 10 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Quadratic Equations textbook chapter.

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What these solutions cover

All 19 questions in Quadratic Equations are solved in the PDF. Here's what's inside, exercise by exercise:

Checking and Forming Quadratic Equations

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
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 dimensions 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…

Solution of a Quadratic Equation by Factorisation

1.
Find the roots of the following quadratic equations by factorisation:
- (i) x² − 3x − 10 = 0
- (ii) 2x² + x − 6 = 0
- (iii) √2x² + 7x + 5√2 = 0
- (iv) 2x² − x + 1/8 = 0
- (v) 100x² − 20x + 1 = 0
2.
Solve the problems given:
- (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 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…
Find two numbers whose sum is 27 and product is 182.
Find two consecutive positive integers, sum of whose squares is 365.
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
A cottage industry produces a certain number of pottery articles in a day. It was observed 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.

Nature of Roots

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
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
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.
Is it possible to design a rectangular park of perimeter 80 m and area 400 m²? If so, find its length and breadth.
5.
Find the roots of the following quadratic equations, if they exist, using the quadratic formula:
- (i) 2x² − 7x + 3 = 0
- (ii) 2x² + x − 4 = 0
- (iii) 4x² + 4√3x + 3 = 0
- (iv) 2x² + x + 4 = 0
The 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.
The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48. Is this situation possible? If so, determine their present ages.
The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2 16/21, find the fraction.
An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore. 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.
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.
A rectangular field is 20 m longer than it is broad. If the area of the field is 8000 m², find the cost of fencing it at Rs 25.50 per metre.

Sample

Worked examples

3 questions from Quadratic Equations, solved step by step.

Solution of a Quadratic Equation by Factorisation

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

Let one number = x.

Then the other number = 27 − x.

Product = x(27 − x) = 182

27x − x² = 182

x² − 27x + 182 = 0

We need two numbers whose product = 182 and sum = 27.

Those numbers are 13 and 14.

x² − 13x − 14x + 182 = 0

x(x − 13) − 14(x − 13) = 0

(x − 13)(x − 14) = 0

x = 13 or x = 14

If x = 13, other number = 27 − 13 = 14.

If x = 14, other number = 27 − 14 = 13.

Verification: 13 + 14 = 27 ✓ and 13 × 14 = 182 ✓

∴ The two numbers are 13 and 14.

Answer: 13 and 14

Nature of Roots

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.

Let breadth = x metres.

Then length = 2x metres.

Area = length × breadth = 2x × x = 2x²

Given: 2x² = 800

x² = 400

x = ±20

Since breadth cannot be negative, x = 20.

Breadth = 20 m, Length = 2 × 20 = 40 m.

This equation (x² = 400) has D = 0² − 4(1)(−400) = 1600 > 0.

So real roots exist, and it is possible to design such a grove.

∴ Length = 40 m, Breadth = 20 m.

Answer: Yes. Length = 40 m, Breadth = 20 m

Solution of a Quadratic Equation by Factorisation

A cottage industry produces a certain number of pottery articles in a day. It was observed 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.

Let the number of articles produced = x.

Cost of each article = Rs (2x + 3).

Total cost = x(2x + 3) = 90

2x² + 3x = 90

2x² + 3x − 90 = 0

We need two numbers whose product = 2 × (−90) = −180 and sum = 3.

Those numbers are 15 and −12.

2x² + 15x − 12x − 90 = 0

x(2x + 15) − 6(2x + 15) = 0

(2x + 15)(x − 6) = 0

x = −15/2 or x = 6

Since number of articles cannot be negative, x = 6.

Cost of each article = 2(6) + 3 = Rs 15.

Verification: 6 × 15 = 90 ✓

∴ Number of articles = 6, Cost of each article = Rs 15.

Answer: 6 articles, Rs 15 each

All 19 questions in Quadratic Equations are solved step by step in the Solutions PDF above.

Board papers

Quadratic Equations in CBSE board papers

Tested in 87 questions across 27 Class 10 board papers (2020–2026). Open a set to see exactly how this chapter is asked:

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Chapter 4 Solutions — Quadratic Equations

Overview

What these solutions cover

Checking and Forming Quadratic Equations

Solution of a Quadratic Equation by Factorisation

Nature of Roots

Worked examples

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

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.

Quadratic Equations in CBSE board papers

More solutions in Mathematics

More NCERT Solutions for Class 10