Quadratic Equations

A polynomial equation of second degree is called a Quadratic Equations.


The standard form of the equation is ax
2 + bx + c = 0 where a,b and c are constants and a≠0.
The exponent of the x should be an integer.

Examples:

1) x
2 + 5x + 10 = 0

Solution : It is a quadratic equation as the degree of the equation is 2.

2) 4a
2 + 5a - 15= 0

Solution : It is a quadratic equation as the degree of the equation is 2.

3) 2/3p
2 - 2p + 20 = 0

Solution : It is a quadratic equation as the degree of the equation is 2.

Example :

x
2 + 2x 1/2 + 4 = 0 is not a quadratic equation as it contains the term x 1/2 , where ½ is not an integer.

The constant ‘a' assuming the value zero reduces the equation to linear form.

When we graph such equation, we get a parabola.
Example : f(x) = x
2 + 6x + 8



The above graph is a upward graph with lowest point known as the
Vertex of the parabola whose coordinates are ( -3, -1).

The points where graph cut the X- axis represent the zeros of the function. The X- coordinates give the
roots of the equation.

Note : Roots of a Quadratic Equation : Let P(x) = 0 , be a quadratic equation, then the zeros of the polynomial P(x) are called the roots of the equation P(x) = 0
Thus x = α is a roots of P(x) = 0 if and only if P(α)= 0

Different forms of the quadratic equations :

1) Standard form :
a2+ bx + c =0
Used to find roots of the equation.
2) Vertex form
a(x-h)2 + k
Used to draw a parabola
3) Factored form :
a(x - x1)(x - x2)= 0
Roots are x1 and x2

There are different methods to factoring the Quadratic Equations.

Check whether the following equations are quadratic or not?

1) 4x
2 - 4 = 0

Solution :
It is a quadratic equation since the highest degree is 2.

2) x - 1/x = x
2

Solution :
x - 1/x = x
2

(x
2 - 1)/x = x 2
∴ x
2 - 1 = x 3 [ cross multiply]

∴ -x
3 + x 2 - 1 = 0

The degree of the equation is 3 so its not a quadratic equation.

3) 5 - x(x - 3) = 0

Solution :
5 - x(x - 3) = 0

5 - x
2 + 3x = 0

The degree of the equation is 2 so its a quadratic equation.
According to the equation, a=-1,b=3 and c=5
the axis of symmetry is x = x-coordinate of vertex.
x-coordinate of vertex = -b/2a
x = 3/(-2 x 1)
x = -3/2
x = -1.5
Equation of axis of symmetry is, x = -1.5

Introduction of Quadratic Equations

Splitting of middle term
By completing the square
Factorization using Quadratic Formula
Vertex form from Quadratic Equation
Finding Axis of Symmetry in Quadratic equation
Solved Problems on Quadratic Equation

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