A nonlinear function that can be written on the standard form

*ax ^{2}+bx+c*, where

*a≠0 ax*,where

^{2}+bx+c*a≠0*

is called a quadratic function.

All quadratic functions has a U-shaped graph called a parabola. The parent quadratic function is

*y=x ^{2}*

This is how the graph looks like

The lowest point of the parabola is called its vertex and its coordinates are,

*x=−b/2a*

The y-coordinate of the vertex is the maximum or minimum value of the function.

When *a > 0, *parabola opens up,

*a < 0*, parabola opens down,

The vertical line that passes through the vertex of the parabola and divides it into two halves is called the axis of symmetry.

The equation of axis of symmetry is

x=−b/2a

The y-intercept of the equation = c, as shown in the figure below.

## SOLUTIONS OF A QUADRATIC EQUATION

The “solutions” to the Quadratic Equation are where it is equal to zero. They are also called “roots”, or sometimes “zeros”

we can use the special Quadratic Formula:

Quadratic Formula:

Just plug in the values of a, b and c, and do the calculations.

The ± means there are TWO answers:

And,

This is how the solutions look like, for example:

In the solution above, is called the Discriminant, because it differentiates between the solutions or nature of Roots.

Nature of Roots:

- when is positive i.e. >0, we get two distinct Real solutions (Roots)
- when it is zero we get just ONE real solution (both answers are the same i.e. coincident)
- when it is negative i.e. <0, we get No Real Roots.