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Quadratic Equations — Class 10 Mathematics

Mathematics · Chapter 4

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Chapter summary

A quadratic equation has the standard form with . This chapter teaches how to solve such equations by factorisation and by the quadratic formula . The expression , called the discriminant, decides the nature of the roots: two distinct real roots if it is positive, two equal real roots if it is zero, and no real roots if it is negative. Many practical problems about areas, speeds and ages reduce to quadratic equations, so mastering these methods is essential.

Standard formSolving by factorisationQuadratic formulaNature of roots (discriminant)Applications

Key ideas & terms

Quadratic equation: An equation of the form .
Discriminant: The quantity that determines the nature of the roots.
Roots: The values of that satisfy the quadratic equation.
Quadratic formula: .

Extra questions & answers

What does the discriminant tell us about the roots?

If the roots are real and distinct; if they are real and equal; if there are no real roots.

Solve by factorisation.

, so or .

What is the standard form of a quadratic equation?

, where and are real numbers and .

Name the methods of solving a quadratic equation.

By factorisation, by completing the square, and by using the quadratic formula.

Write the quadratic formula.

For , .

What is the discriminant, and what does it tell us?

The discriminant is ; if the roots are real and distinct, if real and equal, and if there are no real roots.

Solve by factorisation.

, so or .

Find the nature of the roots of .

, so the equation has no real roots.

For what value of does have equal roots?

Equal roots need : , so .

The product of two consecutive positive integers is 56. Frame and solve the equation.

Let the integers be and : , so and the integers are 7 and 8.

EXERCISE 4.1

1. Check whether the following are quadratic equations :
(i)

On simplifying,

This is of the form , where . Therefore, it is a quadratic equation.

(ii)

On simplifying,

This is a quadratic equation.

(iii)

Expanding both sides,

The squared terms cancel, so the resulting equation is linear. Therefore, it is not a quadratic equation.

(iv)

On expanding,

Therefore, it is a quadratic equation.

(v)

On expanding,

Therefore, it is a quadratic equation.

(vi)

On simplifying,

The squared terms cancel. Therefore, it is not a quadratic equation.

(vii)

On expanding,

The resulting polynomial has degree . Therefore, it is not a quadratic equation.

(viii)

On expanding the right-hand side,

Therefore, it is a quadratic equation.

2. Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is . 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.

Let the breadth be metres. Then its length is metres. Therefore,

Hence, the required quadratic equation is

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

Let the smaller positive integer be . The next integer is . Therefore,

Hence, the required quadratic equation is

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

Let Rohan’s present age be years. His mother’s present age is years. Three years from now, their ages will be and . Therefore,

Hence, the required quadratic equation is

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

Let the train’s usual speed be . Its reduced speed would be . Therefore,

Multiplying by ,

Hence, the required quadratic equation is

EXERCISE 4.2

1. Find the roots of the following quadratic equations by factorisation:
(i)

Factorising,

Thus,

Therefore, the roots are

(ii)

Factorising,

Therefore,

(iii)

Factorising,

Therefore,

(iv)

Multiplying the equation by ,

Therefore, both roots are equal:

(v)

Factorising,

Thus,

Therefore, both roots are equal:

2. Solve the problems given in Example 1.

For the marbles, the equation obtained in Example 1 is

Factorising,

Thus, the two starting numbers are and . Therefore, John and Jivanti had and marbles, in either order.

For the toys, the equation is

Factorising,

Thus, either toys were produced at a cost of ₹ each, or toys were produced at a cost of ₹ each.

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

Let one number be . The other is . Therefore,

Hence, the two numbers are and .

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

Let the smaller integer be . The next integer is . Then

Since the integers are positive, . Hence, the integers are and .

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

Let the base be . Then the altitude is . By the Pythagoras theorem,

The positive value is . Therefore, the base is and the altitude is .

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 more than twice the number of articles produced on that day. If the total cost of production on that day was ₹, find the number of articles produced and the cost of each article.

Let the number of articles be . The cost of each article is ₹. Therefore,

Since the number of articles must be positive, . The cost of each article is

Therefore, articles were produced, and the cost of each article was ₹.

EXERCISE 4.3

1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
(i)

Here, , , and . The discriminant is

Since , the equation has no real roots.

(ii)

Here, , , and . The discriminant is

Therefore, the equation has two equal real roots. The repeated root is

(iii)

Here, , , and . The discriminant is

Therefore, the equation has two distinct real roots:

2. Find the values of for each of the following quadratic equations, so that they have two equal roots.
(i)

For equal roots, the discriminant must be zero:

Thus, or .

(ii)

Writing the equation in standard form,

For it to remain quadratic, . For equal roots,

The value would not give a quadratic equation. Therefore,

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.

Let the breadth be metres. Then the length is metres. The area condition gives

A measurement cannot be negative, so . Therefore, the design is possible. Its breadth is and its length is .

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

Let one friend’s present age be years. The other friend’s age is years. Four years ago, their ages were and . Therefore,

Its discriminant is

Since the discriminant is negative, there are no real ages satisfying the conditions. Therefore, the situation is not possible.

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

Let the length be metres. Since the perimeter is ,

so the breadth is metres. Using the area,

Thus, . Therefore, the design is possible, with length and breadth .

Quadratic Equations — Class 10 Mathematics

CBSE Class 10 community

Chapter summary

Key ideas & terms

Extra questions & answers

What does the discriminant tell us about the roots?

Solve by factorisation.

What is the standard form of a quadratic equation?

Name the methods of solving a quadratic equation.

Write the quadratic formula.

What is the discriminant, and what does it tell us?

Solve by factorisation.

Find the nature of the roots of .

For what value of does have equal roots?

The product of two consecutive positive integers is 56. Frame and solve the equation.

EXERCISE 4.1

1. Check whether the following are quadratic equations :(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

2. Represent the following situations in the form of quadratic equations :(i) The area of a rectangular plot is . 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 . We need to find the integers.

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

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

EXERCISE 4.2

1. Find the roots of the following quadratic equations by factorisation:(i)

(ii)

(iii)

(iv)

(v)

2. Solve the problems given in Example 1.

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

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

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

EXERCISE 4.3

1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:(i)

(ii)

(iii)

2. Find the values of for each of the following quadratic equations, so that they have two equal roots.(i)

(ii)

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.

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

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

1. Check whether the following are quadratic equations :
(i)

2. Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is . 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.

1. Find the roots of the following quadratic equations by factorisation:
(i)

1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
(i)

2. Find the values of for each of the following quadratic equations, so that they have two equal roots.
(i)

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