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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. These mathematical equations function beyond theory because they serve various purposes in real-life scenarios. 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. This section creates the fundamental base required to develop algebraic reasoning abilities together with problem-solving capability. The ability to identify and solve quadratic equations will help to succeed in the examinations and ensure success in their subsequent educational and competitive examination programs.
The NCERT Class 10 Maths Chapter 4 notes cover a brief outline of the chapter on quadratic equations and can be used for revision. The main topics covered in the quadratic equations Class 10 notes give you the properties and roots of quadratic equations. Class 10 Maths chapter 4 notes also cover the basic equations in the chapter. Quadratic equations Class 10 notes PDF download contains all of these topics. Apart from this, the use of NCERT class 10th maths notes provides you with the ability to examine mathematical concepts rigorously. Also, after studying the NCERT Notes, you will have access to short summaries that help you to remember information fast.
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 the 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.
Students must download the notes below for each chapter to ace the topics.
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:
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