#### Study Materials and Revision Notes for Ch 4 Quadratic Equations Class 10th Maths

- Quadratic Equations
- Roots of a Quadratic Equation
- Number Roots
- Methods for solving Quadratic Equation
- Discriminant
- Nature of Roots

**Quadratic Equations:**A quadratic equation is in the form of ax

^{2}+ bx + c = 0 where a, b, c are real numbers and a ≠ 0.

Example: (i) 3x

^{2}+ 4y - 6 = 0

(ii) 2x

^{2}+ 6 = 0

**Roots of a Quadratic Equation**

Let ax² + bx + c = 0, be a quadratic equation. If α is a root of this equation. It means x = α satisfies this equation i.e., aα² + bα + c = 0

**Number of Roots**

A quadratic equation has two roots. The roots may or may not be real.

**Methods For Solving Quadratic Equation**

There are three methods for solving quadratic equation:

(a) By factorization

(b) By completing the square

(c) By Quadratic Formula

**By Factorization**

Only real roots are found by factorization. If we can factorize ax² + bx + c = 0, where a ≠ 0, into a product of two linear factors, then the roots of this quadratic equation can be calculated by equating each factor to zero.

For example:

*x*

^{2}– 3

*x*– 10 = 0

⇒

*x*

^{2}- 5

*x*+ 2

*x*- 10 = 0

⇒

*x*(

*x*- 5) + 2(

*x*- 5) = 0

⇒ (

*x*- 5)(

*x*+ 2) = 0

Roots of this equation are the values for which (

*x*- 5)(

*x*+ 2) = 0

∴

⇒ *x*- 5 = 0 or*x*+ 2 = 0*x*= 5 or

*x*= -2

**By completing the square method**

A quadratic equation can also be solved by the method of completing the square. We will try to convert the given equation in the form of (a+b)

^{2 }or (a-b)

^{2}.

For example:

2

⇒ 2

On dividing both sides of the equation by 2, we get

On adding (7/4)

⇒ (

⇒ (*x*^{2}– 7*x*+ 3 = 0⇒ 2

*x*^{2}– 7*x*= - 3On dividing both sides of the equation by 2, we get

⇒

⇒ *x*^{2}– 7*x*/2 = -3/2*x*^{2}– 2 ×*x*× 7/4 = -3/2On adding (7/4)

^{2}to both sides of equation, we get⇒ (

*x*)^{2 }- 2 ×*x*× 7/4 + (7/4)^{2}= (7/4)^{2}- 3/2*x*- 7/4)

^{2}= 49/16 - 3/2

⇒ (

*x*- 7/4)

^{2}= 25/16

⇒ (

*x*- 7/4) = ± 5/4

⇒

*x*= 7/4 ± 5/4

⇒

*x*= 7/4 + 5/4 or

*x*= 7/4 - 5/4

⇒

*x*= 12/4 or

*x*= 2/4

⇒

*x*= 3 or 1/2

**Quadratic Formula**

To find roots of ax² + bx + c = 0 is given by

We will solve a question using quadratic formula.

4

*x*

^{2}+ 4√3

*x*+ 3 = 0

On comparing this equation with

*ax*

^{2}+

*bx*+

*c*= 0, we get

*a*= 4,

*b*= 4√3 and c = 3

By using quadratic formula, we get

*x*= -

*b*±√

*b*

^{2}- 4

*ac*/2a

⇒

*x*= -4√3±√48-48/8

⇒

*x*= -4√3±0/8

∴

*x*= -√3/2 or

*x*= -√3/2

**Discriminant**

For the quadratic equation ax² + bx + c = 0 the expression is
called the discriminant and denoted by D. Then the roots of the quadratic
equation are given by

**Nature of roots**

The nature of roots for ax² + bx + c = 0 can be determined through discriminant.

(i) If D 0, then roots are real and unequal.

(ii) D=0, then the equation has equal and real roots.

(iii) D<0, then the equation has no real roots