NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations (Ex 4.1) Exercise 4.1

EXERCISE NO: 4.1

1. Check whether the following are quadratic equations:

(i) ${{\left( \text{x + 1} \right)}^{\text{2}}}\text{ = 2}\left( \text{x - 3} \right)$

Ans: We are given an equation: ${{\left( \text{x + 1} \right)}^{\text{2}}}\text{ = 2}\left( \text{x - 3} \right)$

We will simplify the given equation.

${{\left( \text{x + 1} \right)}^{\text{2}}}\text{ = 2}\left( \text{x - 3} \right)$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ + 2x + 1 = 2x - 6}$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ + 1 = - 6}$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ + 7 = 0}$

The equation obtained after simplifying is of the form $\text{a}{{\text{x}}^{\text{2}}}\text{ + bx + c = 0}$.

Therefore, the given equation is a quadratic equation.

(ii) ${{\text{x}}^{\text{2}}}\text{ - 2x = }\left( \text{-2} \right)\left( \text{3 - x} \right)$

Ans: We are given an equation: ${{\text{x}}^{\text{2}}}\text{ - 2x = }\left( \text{-2} \right)\left( \text{3 - x} \right)$

We will simplify the given equation.

${{\text{x}}^{\text{2}}}\text{ - 2x = }\left( \text{-2} \right)\left( \text{3 - x} \right)$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ - 2x = - 6 + 2x}$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ - 4x + 6 =0}$

The equation obtained after simplifying is of the form $\text{a}{{\text{x}}^{\text{2}}}\text{ + bx + c = 0}$.

Therefore, the given equation is a quadratic equation.

(iii) $\left( \text{x - 2} \right)\left( \text{x + 1} \right)\text{ = }\left( \text{x - 1} \right)\left( \text{x + 3} \right)$

Ans: We are given an equation: $\left( \text{x - 2} \right)\left( \text{x + 1} \right)\text{ = }\left( \text{x - 1} \right)\left( \text{x + 3} \right)$

We will simplify the given equation.

$\left( \text{x - 2} \right)\left( \text{x + 1} \right)\text{ = }\left( \text{x - 1} \right)\left( \text{x + 3} \right)$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ - 2x + x - 2 = }{{\text{x}}^{\text{2}}}\text{ - x + 3x - 3}$

$\Rightarrow \text{- 2x + x - 2 = - x + 3x - 3}$

$\Rightarrow \text{- x - 2 = 2x - 3}$

$\Rightarrow \text{3x - 1 = 0}$

This equation obtained is not of the form $\text{a}{{\text{x}}^{\text{2}}}\text{ + bx + c = 0}$.

Therefore, the given equation is not a quadratic equation.

(iv) $\left( \text{x - 3} \right)\left( \text{2x + 1} \right)\text{ = x}\left( \text{x + 5} \right)$

Ans: We are given an equation: $\left( \text{x - 3} \right)\left( \text{2x + 1} \right)\text{ = x}\left( \text{x + 5} \right)$

We will simplify the given equation.

$\left( \text{x - 3} \right)\left( \text{2x + 1} \right)\text{ = x}\left( \text{x + 5} \right)$

$\Rightarrow \text{2}{{\text{x}}^{\text{2}}}\text{- 6x + x - 3 = }{{\text{x}}^{\text{2}}}\text{ + 5x}$

\[\Rightarrow {{\text{x}}^{\text{2}}}\text{- 10x - 3 = 0}\]

The equation obtained after simplifying is of the form $\text{a}{{\text{x}}^{\text{2}}}\text{ + bx + c = 0}$.

Therefore, the given equation is a quadratic equation.

(v) $\left( \text{2x - 1} \right)\left( \text{x - 3} \right)\text{ = }\left( \text{x + 5} \right)\left( \text{x - 1} \right)$

Ans: We are given an equation: $\left( \text{2x - 1} \right)\left( \text{x - 3} \right)\text{ = }\left( \text{x + 5} \right)\left( \text{x - 1} \right)$

We will simplify the given equation.

$\Rightarrow \text{2}{{\text{x}}^{\text{2}}}\text{ - x - 6x + 3 = }{{\text{x}}^{\text{2}}}\text{ + 5x - x - 5}$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ - 7x + 3 = 4x - 5}$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ - 11x + 8 = 0}$

The equation obtained after simplifying is of the form $\text{a}{{\text{x}}^{\text{2}}}\text{ + bx + c = 0}$.

Therefore, the given equation is a quadratic equation.

(vi) ${{\text{x}}^{\text{2}}}\text{ + 3x + 1 = }{{\left( \text{x - 2} \right)}^{\text{2}}}$

Ans: We are given an equation: ${{\text{x}}^{\text{2}}}\text{ + 3x + 1 = }{{\left( \text{x - 2} \right)}^{\text{2}}}$

We will simplify the given equation.

${{\text{x}}^{\text{2}}}\text{ + 3x + 1 = }{{\left( \text{x - 2} \right)}^{\text{2}}}$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ + 3x + 1 = }{{\text{x}}^{\text{2}}}\text{ - 4x + 4}$

$\Rightarrow \text{3x + 1 = - 4x + 4}$

$\Rightarrow \text{7x - 3 = 0}$

This equation obtained is not of the form $\text{a}{{\text{x}}^{\text{2}}}\text{ + bx + c = 0}$.

Therefore, the given equation is not a quadratic equation.

(vii) ${{\left( \text{x + 2} \right)}^{\text{3}}}\text{ = 2x}\left( {{\text{x}}^{\text{2}}}\text{ - 1} \right)$

Ans: We are given an equation: ${{\left( \text{x + 2} \right)}^{\text{3}}}\text{ = 2x}\left( {{\text{x}}^{\text{2}}}\text{ - 1} \right)$

We will simplify the given equation.

${{\left( \text{x + 2} \right)}^{\text{3}}}\text{ = 2x}\left( {{\text{x}}^{\text{2}}}\text{ - 1} \right)$

$\Rightarrow {{\text{x}}^{\text{3}}}\text{ + 6}{{\text{x}}^{\text{2}}}\text{ + 12x + 8 = 2}{{\text{x}}^{\text{3}}}\text{ - 2x}$

$\Rightarrow \text{6}{{\text{x}}^{\text{2}}}\text{ + 14x + 8 = }{{\text{x}}^{\text{3}}}$

$\Rightarrow {{\text{x}}^{\text{3}}}\text{ - 6}{{\text{x}}^{\text{2}}}\text{ - 14x - 8 = 0}$

This equation obtained is not of the form $\text{a}{{\text{x}}^{\text{2}}}\text{ + bx + c = 0}$.

Therefore, the given equation is not a quadratic equation.

(viii) \[{{\text{x}}^{\text{3}}}\text{ - 4}{{\text{x}}^{\text{2}}}\text{ - x + 1 = }{{\left( \text{x - 2} \right)}^{\text{3}}}\]

Ans: We are given an equation: ${{\text{x}}^{\text{3}}}\text{ - 4}{{\text{x}}^{\text{2}}}\text{ - x + 1 = }{{\left( \text{x - 2} \right)}^{\text{3}}}$

We will simplify the given equation.

$\Rightarrow {{\text{x}}^{\text{3}}}\text{ - 4}{{\text{x}}^{\text{2}}}\text{ - x + 1 = }{{\text{x}}^{\text{3}}}\text{ - 6}{{\text{x}}^{\text{2}}}\text{ + 12x - 8}$

$\Rightarrow \text{- 4}{{\text{x}}^{\text{2}}}\text{ - x + 1 = - 6}{{\text{x}}^{\text{2}}}\text{ + 12x - 8}$

$\Rightarrow \text{2}{{\text{x}}^{\text{2}}}\text{ - 13x + 9 =  0}$

This equation obtained is of the form $\text{a}{{\text{x}}^{\text{2}}}\text{ + bx + c = 0}$.

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 $\text{528 }{{\text{m}}^{\text{2}}}$. 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.

Ans: It is given that the area of rectangular plot is $\text{528 }{{\text{m}}^{\text{2}}}$.

Let us assume the breadth of the plot be $\text{b = x m}$.

It is given that the length is one more than twice the breadth.

So, the length will be $\text{l = }\left( \text{2x + 1} \right)\text{ m}$.

The area of the rectangle will be  $\text{A = l  }\!\!\times\!\!\text{  b}$

$\therefore \text{l  }\!\!\times\!\!\text{  b = 528}$

$\Rightarrow \text{x}\left( \text{2x + 1} \right)\text{ = 528}$

$\Rightarrow \text{2}{{\text{x}}^{\text{2}}}\text{ + x - 528 = 0}$

Therefore, the required quadratic equation is $\text{2}{{\text{x}}^{\text{2}}}\text{ + x - 528 = 0}$.

(ii) The product of two consecutive positive integers is $\text{306}$. We need to find the integers.

Ans: It is given that the product of two consecutive positive integers is $\text{306}$.

If we assume the first integer to be $\text{x}$, then the next consecutive integer will be $\left( \text{x+1} \right)$.

So, we can write $\text{x}\left( \text{x + 1} \right)\text{ = 306}$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ + x - 306 = 0}$

Therefore, the required quadratic equation is ${{\text{x}}^{\text{2}}}\text{ + x - 306 = 0}$.

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

Ans: Let us assume Rohan’s age be $\text{x}$.

His mother is $\text{26}$ years older than him.

So, his mother’s age will be $\left( x+26 \right)$

After $\text{3}$ years, Rohan’s age $\text{= x + 3}$

His mother’s age $\text{= x + 26 + 3 = x + 29}$

It is given that after $\text{3}$ years, the product of their ages is $\text{360}$.

$\therefore \left( \text{x + 3} \right)\left( \text{x + 29} \right)\text{ = 360}$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ + 32x + 87 = 360}$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ + 32x - 273 = 0}$

Therefore, the required quadratic equation is ${{\text{x}}^{\text{2}}}\text{ + 32x - 273 = 0}$.

(iv) A train travels a distance of $\text{480}$ $\text{km}$ at a uniform speed. If the speed had been $\text{8}$ $\text{km/h}$ less, then it would have taken $\text{3}$ hours more to cover the same distance. We need to find the speed of the train.

Ans: Let us assume the speed of the train be $\text{x km/h}$.

The train travels a distance of $\text{480 km}$ at a uniform speed.

So, the time taken to travel the given distance $\text{= }\frac{\text{480}}{\text{x}}\text{ hrs}$.

It is given that the train would take $\text{3}$ hours more if the

speed had been $\text{8 km/h}$ less.

Now, the new speed of the train $\text{= }\left( \text{x - 8} \right)\text{ km/h}$.

The new time taken to travel the same distance $\text{= }\left( \frac{\text{480}}{\text{x}}\text{ + 3} \right)\text{ hr}$.

We know that $\text{speed  }\!\!\times\!\!\text{  time = distance}$

$\therefore \left( \text{x - 8} \right)\left( \frac{\text{480}}{\text{x}}\text{ + 3} \right)\text{ = 480}$

$\Rightarrow \text{480 + 3x - }\frac{\text{3840}}{\text{x}}\text{ - 24 = 480}$

$\Rightarrow \text{3x - }\frac{\text{3840}}{\text{x}}\text{ = 24}$

$\Rightarrow \text{3}{{\text{x}}^{\text{2}}}\text{ - 24x - 3840 = 0}$

$\Rightarrow {{\text{x}}^{\text{2}}}\text{ - 8x - 1280 = 0}$

Therefore, the required quadratic equation is ${{\text{x}}^{\text{2}}}\text{ - 8x - 1280 = 0}$.

Ex 4.1 Class 10 - History of Quadratic Equation

The history of the solution of quadratic equations extends across the world for more than 4000 years as class 10 Math chapter 4 exercise 4.1 will tell you. The earliest known records are Babylonian clay tablets from about 1600 BCE where the diagonal of a unit square is given to five decimal places of accuracy. But, as you will know from class 10 quadratic equation exercise 4.1, the use of algebraic symbols only began in the 15th century. Exercise 4.1 Maths class 10 will also inform you that Euclid, a famous Greek mathematician invented the quadratic equations and its solutions while calculating dimensions such as length; breadth and height via geometry. Later Sridharacharya made the calculations of these quadratic equations by completing the square methodology which became more convenient; better known as now as Sridharacharya Formula of Quadratic Equation.

Class 10 Maths Chapter 4 Exercise 4.1 - Concept of Quadratic Equation

The root of the quadratic equation ax2 + bx + c = 0 (a ≠ 0) is given in NCERT class 10 Maths chapter 4 exercise 4.1 by the following formula:

\[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\]

This formula is known as Sridharacharya formula, as you will know from exercise 4.1 class 10 Maths.

Here, the quantity   b2 −4ac   is called the discriminant of the polynomial.

According to Sridharacharya, as mentioned in NCERT Maths class 10 chapter 4 exercise 4.1, every quadratic equation has two roots and the types depend on the value of this discriminant.

If b2−4ac  > 0, the roots of the quadratic equation are real and distinct in nature.

If b2−4ac  < 0, the roots of the quadratic equation are unreal i.e. has a complex solution.

If b2−4ac   = 0, the roots of the quadratic equation are real and equal.

Example 1:

Examine the nature of the roots of the following quadratic equation.

3x2 + 8x + 4  =  0.

Solution:

The given quadratic equation is in the general form:

ax2 + bx + c  =  0

Then, we have a  =  3, b  =  8 and c  =  4.

Therefore: 

b2 - 4ac  =  82 - 4(3)(4)

b2 - 4ac  =  64 - 48

b2 - 4ac  =  16

Here, b2 - 4ac > 0 and also a perfect square.

So, the roots are real and distinct.

Example 2:

Examine the nature of the roots of the following quadratic equation.

5x2 - 4x + 2  =  0

Solution:

The given quadratic equation is in the general form:

ax2 + bx + c  =  0

Then, we have a  =  5, b  =  -4 and c  =  2.

Therefore:

b2 - 4ac  =  (-4)2 - 4(5)(2)

b2 - 4ac  =  16 - 40

b2 - 4ac  =  -24

Here, b2 - 4ac < 0.

So, the roots are imaginary.

Ex 4.1 Class 10 - Solving Techniques of Quadratic Equations

A quadratic equation may be solved either by factorizing the left side( when the right side is zero) or by completing a square on the left side.

Example 1: Solve  6x2 + 11x - 10 = 0

Solution: 6x2 + 11x - 10 = 0

⇒ 6x2 + 15x - 4x - 10 = 0

⇒ 3x(2x + 5) - 2(2x + 5) = 0

⇒ (2x + 5)(3x - 5) = 0

Therefore, either 2x + 5 = 0 or, 3x - 5 = 0

Hence, x = -5/2 or x = ⅔.

Class 10 Chapter 4 Exercise 4.1: Conversion of Non-Quadratic to Quadratic Equation

Sometimes the equations given are not in quadratic order and hence we need to convert it into a quadratic form to solve it.

Example 1: Solve \[2^{2x + 3} + 2^{x + 3} = 1 + 2^{x}\]

Solution: \[2^{2x + 3} + 2^{x + 3} = 1 + 2^{x}\]

or, \[2^{2x} . 2^{3} + 2^{x} . 2^{3} = 1 + 2^{x}\]

or, \[8 . 2^{2x} + 8 . 2^{x} = 1 + 2^{x}\]…(1)

Now, let \[2^{x} = y\] Then, \[y^{2} = (2^{x})^{2} = 2^{2x}\]

∴ from (1) we get,

\[8y^{2} + 8y = 1 + y\]

or, \[8y^{2} + 7y - 1 = 0\]

or, \[8y^{2} - 8y - y - 1 = 0\]

or, \[8y(y + 1) - 1(y + 1) = 0\]

or, \[(y + 1)(8y - 1) = 0\]

∴ y = -1

or ⅛

If, y = -1, then, \[2^{x} = -1\], which is not possible.

If, y = ⅛, then \[2^{x} = \frac{1}{8}\]

or, \[2^{x} = \frac{1}{2^{3}} = 2^{-3}\]

∴ x = -3.

Example 2: Solve \[\sqrt{\frac{x}{1 - x}} + \sqrt{\frac{1 - x}{x}} = \frac{13}{6}\]

Solution: Let us put \[y = \sqrt{\frac{1}{1-x}}\], then \[\frac{1}{y} = \sqrt{\frac{1 - x}{x}}\]

\[\therefore y + \frac{1}{y} = \frac{13}{6} or, \frac{y^{2} + 1}{y} = \frac{13}{6}\]

⇒ 6y2 + 6 = 13y

⇒ 6y2 - 13y + 6 = 0

⇒ 6y2 - 4y - 9y + 6 = 0

⇒ (3y - 2)(2y - 3) = 0

∴ y = ⅔ or, y = 3/2

If y = ⅔ , then \[\sqrt{\frac{x}{1-x}} = \frac{2}{3}\]

\[\Rightarrow \therefore \frac{x}{1 - x} = \frac{4}{9}\]

⇒ 9x = 4 - 4x

⇒ 13x = 4

∴ x = 4/13

If y = 3/2, then \[\sqrt{\frac{x}{1-x}} = \frac{3}{2}\]

\[\Rightarrow \therefore \frac{x}{1 - x} = \frac{9}{4}\]

⇒ 4x = 9 - 9x

⇒ 13x = 9

∴ x = 9/13

Class 10 Maths Ex 4.1 - Sum and Product of Roots of a Quadratic Equation

If α and β be the roots of the quadratic equation ax2 + bx + c = 0 (a ≠ 0) then,

\[\alpha + \beta = -\frac{b}{a}\] is called the sum of roots and \[\alpha \beta = \frac{c}{a}\] is called products of roots of the quadratic equations.

Example 1: If the roots of the equation x2 + px + 7 = 0 are denoted by α and β and α2 + β2 = 22, find the value of p.

Solution: \[\alpha + \beta = \frac{-p}{1} = (-p)\] and \[\alpha \beta = \frac{7}{1} = 7\]

Now, α2 + β2 = 22 (given)

or, (α + β)2 - 2αβ = 22

or, (-p)2 - 2(7) = 22

or, p2 = 22 + 14 = 36

p = ±6

Class 10 Maths Ch 4 Ex 4.1- Problems based on Quadratic Equation

Class 10th Maths Chapter 4 exercise 4.1 has several problems for you to solve. Following are a few of them.

Example 1: In a class test, the sum of Rekha’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: with the help of class 10 Maths exercise 4.1, let us marks in Maths as xx,

Then as per the problem,

Marks in English would be = 30 − x30 − x.

Now if Rekha got 2 marks less, then Maths marks would = (x − 2)(x − 2)

and if she got 3 marks less in English, then English marks would = (30 − x − 3) = (27 − x)(30 − x − 3) = (27 − x)

As per problem, the product is equal to 210 so

(x − 2)(27 − x) = 120(x − 2)(27 − x) = 120

−x2 + 25x + 54 = 210 − x2 + 25x + 54 = 210

or

x2 − 25 x + 156 = 0 x 2 − 25 x +156 = 0

x2 − 90x + 30x − 2700 = 0 x 2 − 90x + 30x − 2700 = 0

or x = 12 or 13

So Rekha’ marks in Maths is 12 or 13 and the English marks are 18 or 17.

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