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Imagine after building your dream house, you need to fence that house, and for that, you need the length and breadth of that house to find the biggest possible area. Quadratic equations can solve this problem. It is an integral part of Mathematics that deals with algebraic equations of degree 2. People use quadratic equations in all parts of life to determine areas and moving object speeds, and make financial predictions while resolving problems throughout physics and engineering. The main purpose of these NCERT Notes of Quadratic Equations class 10 PDF is to provide students with an efficient study material from which they can revise the entire chapter.
After going through the textbook exercises and solutions, students need a type of study material from which they can recall concepts in a shorter time. Quadratic Equations Class 10 Notes are very useful in this regard. In this article about NCERT Class 10 Maths Notes, everything from definitions and properties to detailed notes, formulas, diagrams, and solved examples is fully covered by our subject matter experts at Careers360 to help the students understand the important concepts and feel confident about their studies. These NCERT Class 10 Maths Chapter 4 Notes are made in accordance with the latest CBSE syllabus while keeping it simple, well-structured and understandable. For the syllabus, solutions, and chapter-wise PDFs, head over to this link: NCERT.
Use the link below to download the Quadratic Equations Class 10 Notes PDF for free. After that, you can view the PDF anytime you desire without internet access. It is very useful for revision and last-minute studies.
Quadratic Equations deal with equations of the form
A polynomial with degree 2 is a quadratic polynomial. It is in the form of: f(x) = ax2 + bx + c, where a ≠ 0.
A quadratic polynomial, when equated to a constant (like 0), then the equation becomes a Quadratic Equation, which means f(x) = 0.
The standard form of a Quadratic Equation is: f(x) = ax2 + bx + c = 0, where a, b and c are real numbers and a ≠ 0.
Here, a is called as quadratic coefficient, as it is the coefficient of x2 and b is called as linear coefficient, as it is the coefficient of x, and c is the constant term.
A quadratic equation's roots represent x values that satisfy the given equation.
Let x = α, and α is a real number. If α satisfies the quadratic equation ax2+ bx + c = 0 such that aα2 + bα + c = 0, then α is the root of the Quadratic Equation.
There are three methods to solve Quadratic Equations.
In this method, factorize the equation into two linear factors and equate each factor to zero to find the roots of the equation.
Step 1: Quadratic Equation in the form of ax2 + bx + c = 0.
Step 2: Find the two numbers (let's say p and q), whose sum is equal to 'b' and product is equal to 'a × c'.
Step 3: By factorization, write ax2 + bx + c = 0 as (x + p) (x + q) = 0
For example-
x2- 2x - 15=0
⇒ (x+3)(x-5)=0
So, x + 3 = 0 or, x - 5 = 0
The above values of x are the two roots of the given quadratic equation.
In this method, convert the equation to square form (x + a)2 - b2 = 0 or (x + a)2 = b2 to find the roots.
Example: Solve
Solution:
Step 1: Make sure the coefficient of
In this case, it's already 1.
If not, divide the equation with the coefficient of
Step 2: Move the constant term to the right-hand side:
Step 3: Add the square of half the coefficient of
Half of 6 is 3, and
So,
Step 4: Write the left-hand side as a perfect square, to make the form (x + a)2 = b2
Step 5: Take the square root of both sides:
Step 6: Solve for
Therefore,
In this method, find the roots by using the quadratic formula. The quadratic formula is
where a, b and c are the real numbers and b2 – 4ac is known as the discriminant. In this method, we directly get the roots once we substitute the values in the formula.
Example: Solve
Solution: Here, a = 2, b = -4, c = -6
Apply the quadratic formula:
The nature of the roots of the equation depends upon the value of D, which is called the discriminant.
Value of discriminant |
Number of roots |
D > 0 |
Two distinct real roots |
D = 0 |
Two equal and real roots |
D < 0 |
No real roots |
When the quadratic equation has D > 0, the roots of the equation are distinct and real.
Example: Solve
Solution: Here, a = 1, b = -5, c = 6
D =
It means D > 0.
Now, apply the quadratic formula:
x =
x =
Therefore, the roots are distinct and real.
When the quadratic equation has D = 0, the roots of the equation are equal.
Example: Solve
Solution: Here, a = 1, b = -4, c = 4
D =
It means D = 0.
Now, apply the quadratic formula:
x =
Therefore, x = 2
Thus, the roots are equal, as we get only one root.
When the quadratic equation has D < 0, the roots of the equation are not real (complex).
Example: Solve
Solution: Here, a = 1, b = 2, c = 5
D =
It means D < 0.
Now, apply the quadratic formula:
x =
x =
Thus, the roots are not real and are complex numbers.
Given below are some previous year question answers of various examinations from the NCERT class 10 chapter 4, Quadratic Equations:
Question 1: Find the nature of roots of the equation
Solution:
Given,
For the standard quadratic equation
The sum of the roots is
The product of the roots is
Discriminant
where
Substitute the above-mentioned values in equation (1)
Question 2: Solve the quadratic equation
Solution:
There are two roots of the equation.
Hence, the answer is (
Question 3: The sum of a number and its reciprocal is
Solution:
Let the number be
Hence, the answer is (
All the links to chapter-wise notes for NCERT class 10 maths are given below:
Students must check the NCERT Exemplar solutions for class 10 of Mathematics and Science Subjects.
Students must check the NCERT solutions for class 10 of Mathematics and Science Subjects.
To learn about the NCERT books and syllabus, read the following articles and get a direct link to download them.
A quadratic polynomial, when equated to 0, then the equation becomes a Quadratic Equation, which means f(x) = 0.
The standard form of a Quadratic Equation is: f(x) = ax2 + bx + c = 0, where a, b and c are the real numbers and a ≠ 0.
To recognize a quadratic equation, check:
The methods for solving quadratic equations are as follows:
The standard form of a Quadratic Equation is: f(x) = ax2 + bx + c = 0, where a, b and c are the real numbers and a ≠ 0.
The nature of the roots of the equation depends upon the value of D, which is called the discriminant, where b2 – 4ac is known as the discriminant.
It tells if:
Exam Date:22 July,2025 - 29 July,2025
Exam Date:22 July,2025 - 28 July,2025
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