Topic:

QUADRATIC EQUATION


Quadratic Equation :

An equation of the 2nd degree is called quadratic equation.

Standard or General Form of Quadratic equation in one variable x is ax^2+bx+c=0 ( where a≠0 and a,b,c are real numbers.)


*If a=0 in ax^2+bx+c=0 then it reduces to a linear equation bx+c=0.


*If b=0 in ax^2+bx+c=0 then Equation becomes ax^2+c=0 is called pure Quadratic equation

Discriminant
Discriminant


Graph :

Graph of the quadratic equation is parabolla.

Imporatnce of the co-efficient of x^2 in ax^2+bx+c=0 ( i-e "a")


if a > 0 then the concavity of the curve of Quadratic equation is up-ward parabolla and 

if a<0 then the concavity of the curve of the quadratic equation is dowanward parabolla.

Parabola graph of quadratic Equation
Parabola graph of quadratic Equation


Roots :

Quadratic equation have only two roots.

*If one root of ax^2+bx+c=0 is a+ib then other will be a-ib.

*If one root of ax^2+bx+c=0 is p+√q then other will be p-√q.

*In any quadratic equation ax^2+bx+c=0 if a+b+c=0 then one root will be 1 and other will be c/a.


How many ways to solve an quadratic equation???

 Three ways :

1) factorization

2) quadratic formula

3) completing square method


TYPES OF QUADRATIC EQUATIONS :

 There are total 9 types of quadratic equation 5 are main types and 4 are sub types radical quadratic equation.


What are the application of quadratic farmula?


Symbol of macdonald is made by quadratic concept

Also used in 2D structure Form.


 In daily life we fine speed of any thing, area or any thing place...etc are application of quadratic equation or quadratic formula

To form the parabola

Geometric Represtation :


Geometrica representation of quadratic equation? Geometrically define the roots of Quadratic equation?


 Geometrically, it's curve contains two points which usually intersect the x-axis  and these points are Called as Root of that quadratic equation


What is polynomial? How you relate the quadratic equation to polynomial function?


Quadratic equation is one of the type of  polynomials with degree 2. Polynomials are the general form of all equation with degree n.


 The algebraic expression in which the variable power is positive integer is called polynomial and quadratic eq is also polynomial because here variable power is positive integer and polynomials are funtions


Can we call  quadratic equation is a function?


yes it's algebric  function whose domain and rang is set of all real numbers.

Geometrically we can see the vertical line can only touch the graph of Q.E at one point, so we can say it's function and algebrically for all Domain we have unique range elements.


Some important Questions

Q#1

 if graph of a quadratic equation cuts on x-axis then its roots will be real or complex?


Ans : 

The roots are real


Q#2

Give condition on

ax^2 +bx+c=0 in which case one root is zero and other will be nonzero. Where a, b, c are real numbers and a must be Non zero ???


Ans :

if c=0 then one root will be zero and other will be not zero.

Quadratic Equation
Quadratic Equation