Graph of an equation y = bx + c gives us the straight line and it is termed as the linear equation. But, if we add a term ‘ax2’ to it, the graph changes into a parabola. And, the equation after adding ax2 becomes ax2 + bx + c and that is known as a Quadratic equation.
Quadratic equation is an equation which is of degree 2 and is of the form ax2 + bx + c = 0. Solution of the equation is $x$ = $\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$ where ‘a’, ‘b’ and ‘c’ are the real numbers. Therefore, value of a $\neq$ 0.
3x2 + 8x - 9, 4x2 – 5x + 16 and x2 – 45 are all examples of the quadratic equations.

The values of a, b and c in the equation are termed as the coefficients wherein ‘a’ is the coefficient of x2 and b is the coefficient of x and c is usually called as the constant. If we want to compare one quantity with another, we make use of the inequalities symbols. For example: 2 < 10. 
There are different symbols that we use to express the inequalities, where
> indicates greater than
< indicates less than
$\leq$ indicates less than or equal to
$\geq$ indicates greater than or equal to 

The quadratic inequalities are nothing but the quadratic equation with an inequality sign instead of the equal sign.

There are four different types of inequalities which are which are >, <, $\geq$ or $\leq$.
A quadratic inequality is of either of the following forms-
  1. ax2 + bx + c > 0
  2. ax2 + bx + c < 0
  3. ax2 + bx +c $\geq$ 0
  4. ax2 + bx + c $\leq$ 0
where a, b and c are the coefficients and are the real numbers and ‘a’ can never be equal to zero. That means, it is an inequality in which we have a quadratic polynomial on the left hand side and zero on the right side.

We can never have any other degree except two in case of the quadratic inequality.
We know that a quadratic function can be written in any of the forms mentioned below:
  1. Usual form: f(x) = ax2 + bx + c 
  2. Factor form: f(x) = a (x - p) (x - q)
  3. Complete Square form: f(x) = a(x - r)2 + s
To solve any of the quadratic inequality, the factor form is the easiest to use.

The symbols like < and > are termed as strict inequalities, wherein the symbols like $\leq$ and $\geq$ are termed as the weak inequalities.

There are different rules, that we follow while doing the operations of inequalities as follows:
  1. We can add or subtract any number on both the sides of the inequality given to us.
  2. We can multiply or divide by any positive number to the both sides of the inequality given to us.
  3. We can multiply or divide by any negative number on both sides of the inequality given to us. But, the sign of the inequality changes its  direction after we multiply or divide by a negative number.