Quadratic Equations Class 10th Notes - Free NCERT Class 10 Maths Chapter 4 Notes - Download PDF

Imagine after building your dream house, you need to fench that house anf 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. 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 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. Students can use NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations for conceptual clarity. After completing the textbook exercise, students can also refer to NCERT Exemplar Solutions for Class 10 Maths Chapter 4 Quadratic Equations for more practice purposes.

Quadratic Polynomial

A polynomial with degree 2, is a quadratic polynomial. It is in the form of

f(x) = ax² + bx + c, where a ≠ 0

Quadratic Equation

An algebraic expression of the second degree is called a quadratic equation.

• The standard form of a Quadratic Equation

ax²+bx+c=0

where a, b and c are the real numbers and a≠0

Roots of a Quadratic Equation

Let x = α and α be 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.

As quadratic polynomials have degree two, quadratic equations can have two roots. Thus, the zeros of a quadratic polynomial f(x) =ax²+bx+c is the same as the roots of the quadratic equation ax²+ bx + c= 0.

Methods to Solve Quadratic Equations

There are three methods to solve Quadratic Equations.

1. Factorization Method

In this method, divide 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: By factorization, we 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

$\therefore$ x = - 3 or x = 5

The above values of x are the two roots of the given quadratic equation.

2. Completing the Square Method

In this method, convert the equation in the square form (x + a)² - b² = 0 to find the roots.

Step 1: Quadratic Equation in the standard form ax² + bx + c = 0.

Step 2: Divide both sides by a:

$x^2+\frac{b}{a}x+\frac{c}{a}=0$

Step 3: Transfer the constant to RHS, then add the square of the half of the coefficient of x, i.e. $(\frac{b}{2a}{)}^2$ on both the sides

$\begin{array}{rl}& x^2+\frac{b}{a}x=-\frac{c}{a}\\ & \Rightarrow x^2+2\left(\frac{b}{2a}\right)x+{\left(\frac{b}{2a}\right)}^2=-\frac{c}{a}+{\left(\frac{b}{2a}\right)}^2\end{array}$

Step 4: Write LHS as a perfect square and simplify RHS.

${(x+\frac{b}{2a})}^2=\frac{b^2-4ac}{4a^2}$

Step 5: Square root of both sides.

$x+\frac{b}{2a}=\pm \sqrt{\frac{b^2-4ac}{4a^2}}$

Step 6: Constant terms are shifted to the RHS, and the value of x is calculated as there is no variable at the RHS.

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

3. Quadratic formula method

In this method, find the roots by using the quadratic formula. The quadratic formula is

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

where a, b and c are the real numbers and b² – 4ac is known as a discriminant.

Nature of Roots

The nature of the roots of the equation depends upon the value of D, which is called the discriminant.

NCERT Solutions of Class 10: Subject Wise

NCERT solutions are very useful to students when they attempt to solve the exercise on their own and get stuck on questions or can't understand the logic behind the question. The following links will be very helpful in that cause.

• NCERT Solutions for Class 10 Maths
• NCERT Solutions for Class 10 Science

NCERT Class 10 Exemplar Solutions for Other Subjects:

During the initial stage of the preparation phase, the latest syllabus is very handy. Also, after completing the exercises from the textbooks, students can practice the exemplar exercises and read reference books. Students can use the following links for the above-mentioned purposes.

• NCERT Exemplar Class 10 Maths solutions
• NCERT Exemplar Class 10 Science solutions
• NCERT Books for Class 10
• NCERT Syllabus for Class 10