We at ask-math believe that educational material should be free for everyone. Please use the content of this website for in-depth understanding of the concepts. Additionally, we have created and posted videos on our youtube.

We also offer One to One / Group Tutoring sessions / Homework help for Mathematics from Grade 4th to 12th for algebra, geometry, trigonometry, pre-calculus, and calculus for US, UK, Europe, South east Asia and UAE students.

Affiliations with Schools & Educational institutions are also welcome.

Please reach out to us on [email protected] / Whatsapp +919998367796 / Skype id: anitagovilkar.abhijit

We will be happy to post videos as per your requirements also. Do write to us.

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

The standard form of the equation is ax2 + bx + c = 0 where a,b and c are constants and a≠0.
The exponent of the x should be an integer.

Examples:

1) x2 + 5x + 10 = 0

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

2) 4a2 + 5a - 15= 0

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

3) 2/3p2 - 2p + 20 = 0

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

Example :

x 2 + 2x1/2 + 4 = 0 is not a quadratic equation as it contains the term x1/2, 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) = x2 + 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) 4x2 - 4 = 0

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

2) x - 1/x = x2

Solution :
x - 1/x = x2

(x2 - 1)/x = x2
∴ x2 - 1 = x3 [ cross multiply]

∴ -x3 + x2 - 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 - x2 + 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