Learn how to solve Quadratic Inequalities using similar methods that we use for solving equations.

## Related Topics

- How to Graph Quadratic Functions
- How to Solve a Quadratic Equation
- How to Graph Quadratic inequalities

## Step by step guide to solve Solving Quadratic Inequalities

- A quadratic inequality is one that can be written in one of the following standard forms:

\(ax^2+bx+c>0, ax^2+bx+c<0, ax^2+bx+c≥0, ax^2+bx+c≤0\) - Solving a quadratic inequality is like solving equations. We need to find solutions.

### Solving Quadratic Inequalities – Example 1:

Solve quadratic inequality. \(x^2-6x+8>0\)

**Solution:**

Factor: \(x^2-6x+8>0→(x-2)(x-4)>0\)

Then the solution could be \(x<2\) or \(x>4\).

### Solving Quadratic Inequalities – Example 2:

Solve quadratic inequality. \(x^2-7x+10≥0\)

**Solution:**

Factor: \(x^2-7x+10≥0→(x-2)(x-5)≥0\). \(2\) and \(5\) are the solutions. Now, the solution could be \(x<2\) or \(x=2\) and \(x=5\) or \(x>5\).

### Solving Quadratic Inequalities – Example 3:

Solve quadratic inequality. \(- x^2-5x+6>0\)

**Solution:**

Factor: \(- x^2-5x+6>0→-(x-1)(x+6)>0\)

Multiply both sides by \(-1: (-(x-1)(x+6))(-1)>0(-1)→(x-1)(x+6)<0\) Then the solution could be \(-6x\) and \(x>1\). Choose a value between \(-1\) and \(6\) and check. Let’s try \(0\). Then: \(- 0^2-5(0)+6>0→6>0\). This is true! So, the answer is: \(-6<x<1\)

### Solving Quadratic Inequalities – Example 4:

Solve quadratic inequality. \(x^2-3x-10≥0\)

**Solution:**

Factor: \(x^2-3x-10≥0→(x+2)(x-5)≥0. -2\) and \(5\) are the solutions. Now, the solution could be \(-2≤x≤5\) or \(-6≥x\) and \(x≥1\). Let’s choose zero to check:

\(0^2-3(0)-10≥0→-10≥0\), which is not true. So, \(-6≥x\) and \(x≥1\)

## Exercises for Solving Quadratic Inequalities

### Solve each quadratic inequality.

- \(\color{blue}{x^2+7x+10<0}\)
- \(\color{blue}{ x^2+9x+20>0}\)
- \(\color{blue}{x^2-8x+16>0}\)
- \(\color{blue}{ x^2-8x+12≤0}\)
- \(\color{blue}{ x^2-11x+30≤0}\)
- \(\color{blue}{ x^2-12x+27≥0}\)

### Download Solving Quadratic Inequalities Worksheet

- \(\color{blue}{-5<x<-2}\)
- \(\color{blue}{x<-5 \ or \ x>-4}\)
- \(\color{blue}{x<4 \ or \ x>4}\)
- \(\color{blue}{2≤x≤6}\)
- \(\color{blue}{5≤x≤6}\)
- \(\color{blue}{x≤3 \ or \ x≥9}\)