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.
The quadratic inequality is very much similar to a quadratic equation except that there is an inequality sign in place of equal to sign. Following are few examples of the quadratic inequalities:
  • 3x2 + 5x - 1 < 0. This is considered as a quadratic inequality as it has degree 2 and 'a' = 3 which is not equal to zero and it has an inequality sign also.
  • 5x2 + 1 > 0. This is also a quadratic inequality as the highest degree is 2 and value of a is 5. It does not have any x term but still it is termed as a quadratic inequality.
A quadratic inequality should contain the following parameters:
  1. The degree of the quadratic inequality should be two.
  2. The coefficient of x2 should not be equal to zero.
  3. There should be any inequality sign in the expression.

The expressions 4x - y3 < 0, 7x4 + 6x - 9 < 0, 3m2 - 6m + 18 = 0 are not considered to be the quadratic inequalities. In the first expression, highest degree is three. Similarly, in the second expression, the highest degree is 4 and in the last expression, there is no inequality sign present.

We know that the graph of a quadratic expression is a parabola. Graphing a quadratic inequality is similar to graphing a linear inequality. The only difference is instead of a line, we graph a parabola. When we have a quadratic inequality that is of the form y > ax2 + bx + c, then it actually represents the region of the plane which is bounded by the graph of its parabola. To graph a quadratic inequality, we should follow some steps.

Steps for graphing quadratic inequalities:
  1. The first thing we need to check while graphing a quadratic inequality is about the boundary line, whether it is dashed or solid. If we have < or > inequality signs, then the boundary line is dashed and if we have $\leq$ or $\geq$ inequality signs, then the boundary is solid.
  2. The next step is to take out the inequality and plug in the equal to sign in place of that which results in getting a quadratic equation.
  3. In the next step, we find out the values of a, b and c. 
  4. The next thing we do is check the sign of the coefficient of x2, i.e. the sign of 'a'. If it is positive, then the parabola points upwards and if 'a' is negative that means parabola is facing or pointing downwards. For example: Consider an expression 3x2 + 8x - 9 > 0. Here, 'a' is 3 which is positive. Hence, the parabola will point upwards.
  5. Then, we find out the axis of symmetry which is the line that cuts the parabola into half. The axis of symmetry is equal to $x$ = $\frac{-b}{2a}$.
  6. Next step is to find the vertex of the parabola which is given as $[\frac{-b}{2a}, f(\frac{-b}{2a})]$
  7. Now, we have the vertex and we need the other points of the parabola, which can be calculated by plugging different values of x and finding the simultaneous values of y.We can plug different values of x (positives and negatives)
  8. We connect all our points to form a parabola which is the graph of our quadratic equation.
  9. The next step is to put back the inequality present in the equation. To find out which region should we shade, we take test points. Usually we take the point of origin and plug in the inequality. If it satisfies, then that is the area we need to shade. We can take two or three test points to shade the required region. 
We have to follow the following steps to solve a given inequality of order two algebraically:
  1. First, find the points or roots of the quadratic equation formed from the given inequality.
  2. In between the points so obtained, the intervals obtained are either Greater than zero or Less than zero.
  3. Then, any value is picked that can be tested to find out if it is greater or less than zero.

Solved Examples

Question 1: Solve x2 - x - 6 < 0
Solution:
Let us solve x2 - x - 6 = 0.
x2 - 3x + 2x - 6 = 0
(x - 3) (x + 2) = 0
x = 3   or    x = - 2
Thus, in between -2 and +3, the value of the function will either be always greater than zero or will always be less than zero.
Now, let us take a test value between -2 and 3 say '0'.
x2 - x - 6 at 0
0 - 0 - 6 = -6 < 0
So, in between -2 and 3, the function will always be negative. That is, less than zero.
So, the solution to x2 - x - 6 < 0 is the interval (-2, 3).

Question 2: Solve - (x)2 + 4 < 0
Solution:
First, consider - x2 + 4 = 0.
- x2 + 4 = 0
=> x2 - 4 = 0
=> (x + 2) (x - 2) = 0
=> x = -2    or    x = 2
Thus, three intervals are gained, x < -2, -2 < x < 2, x > 2.
Now, we see on which intervals, the graph lies below x-axis. But, since the function is negatively quadratic, the graph is an upside down parabola. This implies, the graph is high (above the axis) in the middle, and low (below the axis) on the ends.
Quadratic Inequalities
So, the solution is clearly x < -2 or x > 2

Question 3: Solve x2 + x + 1 < 0.
Solution:
First, find the solution to x2 + x + 1 = 0

$x$ = $\frac{-1\pm \sqrt{1^{2}-4(1)(1)}}{2 \times 1}$

= $\frac{-1\pm \sqrt{1-4}}{2}$

= $\frac{-1\pm \sqrt{-3}}{2}$

Since we have a negative inside the root, there must be no x-intercepts.
This implies that the graph of this function must be always above the x-axis or else always below the x-axis as it can never touch or cross the x- axis.
But, y = x2 + x + 1 is a positive function. So, the parabola is right side up as shown below.
Quadratic Inequalities Examples
So, the given function is always greater than zero. Thus, the solution is all x.