Solving quadratic inequalities can be a bit daunting unless you think logically about it.

To be able to solve these, you will need to be able to solve quadratic equations ( It doesn't matter if you factorise or use the formula) and sketch quadratic graphs.

Let's look at a couple of examples.

**Example 1: Solve the inequality x ^{2} - 2x - 3 ≥ 0**

Step 1: Imagine that this is an equation and solve it (use either factorisation or the formula)

**x = -1 and x = 3**

You should know that if we draw the graph for y = x^{2} - 2x - 3. These two points are where the graph crosses the x axis.

Step 2: Sketch the quadratic graph

Step 3: Identify the parts of the graph that satisfy the inequality.

We have the inequality x^{2} - 2x - 3 ≥ 0

Since we are looking for the points that are greater than or equal to 0, we need to highlight the following sections.

We can see that there are two sections highlighted in red. This means we need two inequalities to describe the solutions.

**x ≤ -1 and x ≥ 3**

**Example 2: Solve the inequality x ^{2} - x - 20 < 0**

Step 1: Imagine that this is an equation and solve it (use either factorisation or the formula)

**x = -4 and x = 5**

You should know that if we draw the graph for y = x^{2} - x - 20. These two points are where the graph crosses the x axis.

Step 2: Sketch the quadratic graph

Step 3: Identify the parts of the graph that satisfy the inequality.

We have the inequality x^{2} - x - 20 < 0

Since we are looking for the points that are less than 0, we need to highlight the following section.

We can see that there is only one section highlighted in red. This means we only need one inequality t describe the solutions.

**-4 < x < 5**