**A polynomial of degree 2 is called Quadratic Equation.**

**For example:**

- 2x
^{2}-5x+1=0 - x
^{2}-5=0

**A general form of quadratic equation is ax**

^{2}+bx+c=0
Where a, b, c all belong to real numbers

Now if you compare the equations I and II with the general form.

- a=2, b=-5 and c=1
- a=1, b=0 and c=-5

### Zeros or the solutions of Quadratic Equation

The real value of x for which the value of the P(x) = ax

^{2}+bx+c becomes zero is known as the root of
the quadratic equation ax

^{2}+bx+c =0.- Determine whether 3and 4 are the zeros of the polynomial P(x) =x
^{2}-7x+12

**Solution:**

P (3) = 3

^{2}-7.3+12
= 9-21+12

= -12+12

= 0

P (4) = 42-7.4+12

= 16-28+12

= -12+12

= 0

Therefore 3 and 4 are the zeros or the solutions or the roots of the polynomial P(x)=x

^{2}-7x+12### Kinds of a Quadratic Equation

There are two types of quadratic equations

- Pure quadratic equation
- Adfected quadratic equation

###
__Pure quadratic equation:__

__Pure quadratic equation:__

An equation of the form ax

^{2}+c=0 , a ≠0 is known as the pure quadratic equation .It means that the
quadratic equation ax

^{2}+bx+c =0, having no term containing single power of x, known as pure
quadratic equation . Clearly in a quadratic equation a ≠0 and b=0.

For example

- x
^{2}-4=0 - 3/8x
^{2}=5

###
__Adfected quadratic equation:__

__Adfected quadratic equation:__

A quadratic equation of the form ax

^{2}+bx+c =0, a ≠0 is known adfected quadratic equation or general
quadratic equation. An adfected quadratic equation has also a term containing single power of x .In

adfected quadratic equation

ax

^{2}+bx+c =0, a ≠0, b ≠0**For example:**

- x
^{2}-5x+11=0 - 0.3x
^{2}+17x-2.3=0

## Solving a pure quadratic equation

There are two methods for solving a pure quadratic equation of the form ax2+c=0:

- By square root
- By factorization

###
** ** To solve a quadratic equation by square root

ax

^{2}+c=0 is a pure quadratic equation. To solve it , bring the constant term the RHS (right hand side)
and divide both side by a, coefficient of x2 and take the square root.

For example :

4x

^{2}-49=0
4x

^{2}=49
x

^{2}=49/4
x =

__+__7/2### To solve a pure quadratic equation by factorization

Bring the equation ax

^{2}+c=0 in the form p^{2}-q^{2}=0. Use the p^{2}-q^{2}=(p+q)(p-q). Equate each factor to
zero and find the values of x in each case, the two values of x so obtained are roots of the equation

ax

^{2}+c=0
For example :

16x

^{2}-25=0
(4x)

^{2}-(5)^{2}= 0
(4x+5) (4x-5) =0

4x-5 =0

x= 5/4

4x+5 =0

x =-5/4

## Solving an Adfected Quadratic Equation

There are two methods for solving an adfected quadratic equation ax

^{2}+ bx+c=0, a ≠0, b ≠0**.**

**(i) By factorization (ii) By completing square**

###
**To solve the quadratic equation **ax^{2} + bx+c=0 **by the method factorization**

In this method the middle term (i.e term containing single power of x) is broken into two suitable

parts so that the factors are formed.

**Example:**

**8x**

*solve.*^{2}+ 7x – 15 = 0

**8x**

*Solution:*^{2}+7x- 15 =0

Or 8x

^{2}-8x + 15x-15=0
Or 8x (x-1) +15(x-1) =0

Or (8x +15)(x-1)=0

Therefore 8x+15=0

8x=-15

x= -15/8

And x-1 =0

x-1

Hence -15/8 and 1 are the required roots.

###
To solve ax^{2} + bx+ c=0 by completion of square

The famous Indian mathematician ShreedharAcharya had invented a formula for solving the

quadratic equation ax

^{2}+bx+c=o.
If the equation ax

^{2}+bx +c =0 has roots α and β, then
and

Where a = coefficient of x

^{2}^{}

b = coefficient of x,

c = constant term