Quadratic Equations

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

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