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Here students will get NCERT Solutions For Quadratic Equation class 10 which are created by expert team at Careers360 keeping in mind of latest syllabus of CBSE 2023-24. These solutions are very helpful for students to understand the basic concepts in a better way as these cover all the topics mentioned in syllabus comprehensive also include step by step explanation of each problem. When we equate the quadratic polynomial to zero, then we get a quadratic equation. NCERT Solutions for Class 10 Maths will help students to get indept understanding of the concepts and thus help students to strategise their preparation. They must complete the NCERT Class 10 Maths syllabus to the earliest so that they can revise in a strategic way. Each exercises of Quadratic Equation Class 10 are solved here in detailled manner.
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Understanding quadratic equations chapter 4 maths class 10 is crucial as they appear in various real-life scenarios. Students must focus on mastering this chapter in the 2023-24 CBSE Syllabus to excel in Class 10 Math exams. NCERT solutions are valuable tools for comprehension and self-assessment. Regular practice with these solutions aids in addressing weaknesses. In mathematics, answers are either right or wrong, making concentration essential for achieving full marks.
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Quadratic Equation:
f(x) = ax^{2} + bx + c, Where a, b, c ∈ R and a ≠ 0.
Quadratic Formula - Roots:
(α, β) = [-b ± (b^{2} - 4ac)]/2a
Roots of Quadratic Equation:
x = [-b ± D]/2a
D = b^{2} - 4ac
Nature of Roots of Quadratic Equation:
D > 0: Roots are real and distinct.
D = 0: Roots are real and equal.
D < 0: Roots are imaginary.
The sum of Roots:
S = α + β = -b/a
Product of Roots:
P = αβ = c/a
Quadratic Equation in terms of Roots:
x^{2} - (α + β)x + αβ = 0
Common Roots of Quadratic Equations:
One common root: (b_{1}c_{2} - b_{2}c_{1}) / (c_{1}a_{2} - c_{2}a_{1}) = (c_{1}a_{2} - c_{2}a_{1}) / (a_{1}b_{2} - a_{2}b_{1})
Both roots common: a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}
Quadratic Equation Simplified Form:
ax^{2} + bx + c = 0 or [(x + b/2a)^{2} -D/(4a^{2})]
Extreme Values of Quadratic Equation:
If a > 0, minimum value = (4ac - b^{2})/4a at x = -b/(2a)
If a < 0, maximum value = (4ac - b^{2})/4a at x = -b/(2a)
The sum of Roots of Cubic Equation:
If α, β, γ are roots of the cubic equation: ax^{3} + bx^{2} + cx + d = 0
α + β + γ = -b/a.
αβ + βγ + γα = c/a.
αβγ = -d/a.
Free download NCERT Solutions For Class 10 Maths Chapter 4 Quadratic Equations PDF for CBSE Exam.
Quadratic Equation Class 10 Excercise: 4.1
Q1 (i) Check whether the following are quadratic equations :
We have L.H.S.
Therefore, can be written as:
i.e.,
Or
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q1 (ii) Check whether the following are quadratic equations :
Given equation can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q1 (iii) Check whether the following are quadratic equations :
L.H.S. can be written as:
and R.H.S can be written as:
i.e.,
The equation is of the type: .
Hence, the given equation is not a quadratic equation since a=0.
Q1 (iv) Check whether the following are quadratic equations :
L.H.S. can be written as:
and R.H.S can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q1 (v) Check whether the following are quadratic equations :
L.H.S. can be written as:
and R.H.S can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q1 (vi) Check whether the following are quadratic equations :
L.H.S.
and R.H.S can be written as:
i.e.,
This equation is NOT of type: .
Here a=0, hence, the given equation is not a quadratic equation.
Q1 (vii) Check whether the following are quadratic equations :
L.H.S. can be written as:
and R.H.S can be written as:
i.e.,
This equation is NOT of type: .
Hence, the given equation is not a quadratic equation.
Q1 (viii) Check whether the following are quadratic equations :
L.H.S. ,
and R.H.S can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q2 (i) Represent the following situations in the form of quadratic equations : The area of a rectangular plot is . 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.
Given the area of a rectangular plot is .
Let the breadth of the plot be .
Then, the length of the plot will be: .
Therefore the area will be:
which is equal to the given plot area .
Hence, the length and breadth of the plot will satisfy the equation
Q2 (ii) Represent the following situations in the form of quadratic equations : The product of two consecutive positive integers is 306. We need to find the integers.
Given the product of two consecutive integers is
Let two consecutive integers be and .
Then, their product will be:
Or .
Hence, the two consecutive integers will satisfy this quadratic equation .
Q2 (iii) Represent the following situations in the form of quadratic equations: Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.
Let the age of Rohan be years.
Then his mother age will be: years.
After three years,
Rohan's age will be years and his mother age will be years.
Then according to question,
The product of their ages 3 years from now will be:
Or
Hence, the age of Rohan satisfies the quadratic equation .
Q2 (iv) Represent the following situations in the form of quadratic equations : 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.
Let the speed of the train be km/h.
The distance to be covered by the train is .
The time taken will be
If the speed had been less, the time taken would be: .
Now, according to question
Dividing by 3 on both the side
Hence, the speed of the train satisfies the quadratic equation
Quadratic Equation Class 10 Excercise: 4.2
Q1 (i) Find the roots of the following quadratic equations by factorization:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are .
Q1 (ii) Find the roots of the following quadratic equations by factorization:
Given the quadratic equation:
Factorisation gives,
Hence, the roots of the given quadratic equation are
Q1 (iii) Find the roots of the following quadratic equations by factorization:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are
Q1 (iv) Find the roots of the following quadratic equations by factorization:
Given the quadratic equation:
Solving the quadratic equations, we get
Factorization gives,
Hence, the roots of the given quadratic equation are
Q1 (v) Find the roots of the following quadratic equations by factorization:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are
.
Q2 Solve the problems given in Example 1. (i) (ii)
From Example 1 we get:
Equations:
(i)
Solving by factorization method:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are .
Therefore, John and Jivanti have 36 and 9 marbles respectively in the beginning.
(ii)
Solving by factorization method:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are .
Therefore, the number of toys on that day was
Q3 Find two numbers whose sum is 27 and the product is 182.
Let two numbers be x and y .
Then, their sum will be equal to 27 and the product equals 182.
...............................(1)
.................................(2)
From equation (2) we have:
Then putting the value of y in equation (1), we get
Solving this equation:
Hence, the two required numbers are .
Q4 Find two consecutive positive integers, the sum of whose squares is 365.
Let the two consecutive integers be
Then the sum of the squares is 365.
.
Hence, the two consecutive integers are .
Let the length of the base of the triangle be .
Then, the altitude length will be: .
Given if hypotenuse is .
Applying the Pythagoras theorem; we get
So,
Or
But, the length of the base cannot be negative.
Hence the base length will be .
Therefore, we have
Altitude length and Base length
Let the number of articles produced in a day
The cost of production of each article will be
Given the total production on that day was .
Hence we have the equation;
But, x cannot be negative as it is the number of articles.
Therefore, and the cost of each article
Hence, the number of articles is 6 and the cost of each article is Rs.15.
Class 10 Maths Chapter 4 Quadratic Equations Excercise: 4.3
Q1 (i) Find the roots of the following quadratic equations, if they exist, by the method of completing the square
Given equation:
On dividing both sides of the equation by 2, we obtain
Q1 (ii) Find the roots of the following quadratic equations, if they exist, by the method of completing the square
Given equation:
On dividing both sides of the equation by 2, we obtain
Adding and subtracting in the equation, we get
Q1 (iii) Find the roots of the following quadratic equations, if they exist, by the method of completing the square
Given equation:
On dividing both sides of the equation by 4, we obtain
Adding and subtracting in the equation, we get
Hence there are the same roots and equal:
Q2 (iv) Find the roots of the following quadratic equations, if they exist, by the method of completing the square
Given equation:
On dividing both sides of the equation by 2, we obtain
Adding and subtracting in the equation, we get
Here the real roots do not exist (in the higher studies we will study how to find the root of such equations).
Q2 Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.
(i)
The general form of a quadratic equation is : , where a, b, and c are arbitrary constants.
Hence on comparing the given equation with the general form, we get
And the quadratic formula for finding the roots is:
Substituting the values in the quadratic formula, we obtain
Therefore, the real roots are:
(ii)
The general form of a quadratic equation is : , where a, b, and c are arbitrary constants.
Hence on comparing the given equation with the general form, we get
And the quadratic formula for finding the roots is:
Substituting the values in the quadratic formula, we obtain
Therefore, the real roots are:
(iii)
The general form of a quadratic equation is : , where a, b, and c are arbitrary constants.
Hence on comparing the given equation with the general form, we get
And the quadratic formula for finding the roots is:
Substituting the values in the quadratic formula, we obtain
Therefore, the real roots are:
(iv)
The general form of a quadratic equation is : , where a, b, and c are arbitrary constants.
Hence on comparing the given equation with the general form, we get
And the quadratic formula for finding the roots is:
Substituting the values in the quadratic formula, we obtain
Here the term inside the root is negative
Therefore there are no real roots for the given equation.
Q3 (i) Find the roots of the following equations:
Given equation:
So, simplifying it,
Comparing with the general form of the quadratic equation: , we get
Now, applying the quadratic formula to find the roots:
Therefore, the roots are
Q3 (ii) Find the roots of the following equations:
Given equation:
So, simplifying it,
or
Can be written as:
Hence the roots of the given equation are:
Let the present age of Rehman be years.
Then, 3 years ago, his age was years.
and 5 years later, his age will be years.
Then according to the question we have,
Simplifying it to get the quadratic equation:
Hence the roots are:
However, age cannot be negative
Therefore, Rehman is 7 years old in the present.
Let the marks obtained in Mathematics be 'm' then, the marks obtain in English will be '30-m'.
Then according to the question:
Simplifying to get the quadratic equation:
Solving by the factorizing method:
We have two situations when,
The marks obtained in Mathematics is 12, then marks in English will be 30-12 = 18.
Or,
The marks obtained in Mathematics is 13, then marks in English will be 30-13 = 17.
Let the shorter side of the rectangle be x m.
Then, the larger side of the rectangle wil be .
Diagonal of the rectangle:
It is given that the diagonal of the rectangle is 60m more than the shorter side.
Therefore,
Solving by the factorizing method:
Hence, the roots are:
But the side cannot be negative.
Hence the length of the shorter side will be: 90 m
and the length of the larger side will be
Given the difference of squares of two numbers is 180.
Let the larger number be 'x' and the smaller number be 'y'.
Then, according to the question:
and
On solving these two equations:
Solving by the factorizing method:
As the negative value of x is not satisfied in the equation:
Hence, the larger number will be 18 and a smaller number can be found by,
putting x = 18, we obtain
.
Therefore, the numbers are or .
Let the speed of the train be
Then, time taken to cover will be:
According to the question,
Making it a quadratic equation.
Now, solving by the factorizing method:
However, the speed cannot be negative hence,
The speed of the train is .
Let the time taken by the smaller pipe to fill the tank be
Then, the time taken by the larger pipe will be: .
The fraction of the tank filled by a smaller pipe in 1 hour:
The fraction of the tank filled by the larger pipe in 1 hour.
Given that two water taps together can fill a tank in hours.
Therefore,
Making it a quadratic equation:
Hence the roots are
As time is taken cannot be negative:
Therefore, time is taken individually by the smaller pipe and the larger pipe will be and hours respectively.
Let the average speed of the passenger train be .
Given the average speed of the express train
also given that the time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance.
Therefore,
Can be written as quadratic form:
Roots are:
As the speed cannot be negative.
Therefore, the speed of the passenger train will be and
The speed of the express train will be .
Let the sides of the squares be . (NOTE: length are in meters)
And the perimeters will be: respectively.
Areas respectively.
It is given that,
.................................(1)
.................................(2)
Solving both equations:
or putting in equation (1), we obtain
Solving by the factorizing method:
Here the roots are:
As the sides of a square cannot be negative.
Therefore, the sides of the squares are and .
Quadratic Equation Class 10 Excercise: 4.4
For a quadratic equation, the value of discriminant determines the nature of roots and is equal to:
If D>0 then roots are distinct and real.
If D<0 then no real roots.
If D= 0 then there exists two equal real roots.
Given the quadratic equation, .
Comparing with general to get the values of a,b,c.
Finding the discriminant:
Here D is negative hence there are no real roots possible for the given equation.
Q1 (ii) Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
Here the value of discriminant =0, which implies that roots exist and the roots are equal.
The roots are given by the formula
So the roots are
The value of the discriminant
The discriminant > 0. Therefore the given quadratic equation has two distinct real root
roots are
So the roots are
For two equal roots for the quadratic equation:
The value of the discriminant .
Given equation:
Comparing and getting the values of a,b, and, c.
The value of
Or,
Q2 (ii) Find the values of k for each of the following quadratic equations so that they have two equal roots
For two equal roots for the quadratic equation:
The value of the discriminant .
Given equation:
Can be written as:
Comparing and getting the values of a,b, and, c.
The value of
But is NOT possible because it will not satisfy the given equation.
Hence the only value of is 6 to get two equal roots.
Let the breadth of mango grove be .
Then the length of mango grove will be .
And the area will be:
Which will be equal to according to question.
Comparing to get the values of .
Finding the discriminant value:
Here,
Therefore, the equation will have real roots.
And hence finding the dimensions:
As negative value is not possible, hence the value of breadth of mango grove will be 20m.
And the length of mango grove will be:
Let the age of one friend be
and the age of another friend will be:
4 years ago, their ages were, and .
According to the question, the product of their ages in years was 48.
or
Now, comparing to get the values of .
Discriminant value
As .
Therefore, there are no real roots possible for this given equation and hence,
This situation is NOT possible.
Let us assume the length and breadth of the park be respectively.
Then, the perimeter will be
The area of the park is:
Given :
Comparing to get the values of a, b and c.
The value of the discriminant
As .
Therefore, this equation will have two equal roots.
And hence the roots will be:
Therefore, the length of the park,
and breadth of the park .
Representation of situation in a quadratic equation.
Checking if an equation is quadratic or not.
Solving a quadratic equation using factorization/ roots of quadratic solutions.
The solution of the quadratic equation using completing the square method.
Solving a quadratic equation using the Sridharacharya formula.
Product of roots in a quadratic equation.
Sum of roots in a quadratic equation.
There are 4 exercise in Class 10 Maths chapter 4. Get the exercise wise solutions from the following links.
Expertly Crafted: Skilled teachers at Careers360 create these NCERT Solutions with great care.
Accuracy Guaranteed: Class 10 maths chapter 4 solutions are entirely correct, making them perfect for students getting ready for their CBSE board exams.
Thorough Explanation: Even the smallest details are explained to help students feel more confident when facing other competitive exams.
Step-by-Step Answers: Class 10 maths ncert chapter 4 solutions to the textbook exercises are provided step by step. This helps students not only get the right final answers but also understand each part of the process, which can lead to better scores.
Chapter No. | Chapter Name |
Chapter 1 | |
Chapter 2 | |
Chapter 3 | |
Chapter 4 | Quadratic Equations |
Chapter 5 | |
Chapter 6 | |
Chapter 7 | |
Chapter 8 | |
Chapter 9 | |
Chapter 10 | |
Chapter 11 | |
Chapter 12 | |
Chapter 13 | |
Chapter 14 | |
Chapter 15 |
First of all list down all the questions in which you need assistance and go through the NCERT Solutions For Class 10 Maths Chapter 4 Quadratic Equations of that particular question.
When you complete the first step then your next target should be previous papers. You can pick past year papers and practice them thoroughly.
Once you complete NCERTs and previous year papers, try to solve the questions of that particular chapter from different state board books.
Class 10 maths chapter 4 solutions required multiple concepts including what is an equation, how to get the solution of an equation, methods of quadratic equation solution, the factoring method for solving a quadratic equation, the quadratic formula for solving a quadratic equation, the nature of roots of a quadratic equation, graphs of quadratic equations, and many.
Chapter 4 quadratic equation class 10 ncert solutions contain multiple topics like Quadratic equations, Solutions of a quadratic equation, Discriminant of a quadratic equation, Factoring method for solving a quadratic equation, Quadratic formula for solving a quadratic equation, Nature of roots of a quadratic equation, and Graphs of quadratic equations.
Students can visit the NCERT official website. There they can find options to select class, subject, and book title. Using these inputs students can download NCERT chapter 4 maths class 10 which is quadratic equations class 10 pdf. Students can also find solutions to these NCERTs from Careers360's official website
There are multiple methods to solve a quadratic equation including the factoring method, quadratic formula method, graphical method, and many others. These methods depend on the type of quadratic equation so first to know what is quadratic polynomial type and then apply the above methods to find a quadratic polynomial solution.
To determine whether an equation is a quadratic equation, you can look for two conditions: firstly, the highest power of the variable is 2 and secondly, the coefficient of the x^2 term (that is, the "a" in the equation) is not equal to zero. These two are some examples of quadratic equations 2x^2+3x+5 and x^2+5x+6.
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