How to Solve Quadratic Inequalities

Solving quadratic inequalities is a fundamental skill in algebra that helps you determine the range of values that satisfy a quadratic expression. This guide will walk you through the step-by-step process of solving quadratic inequalities effectively.

Quadratic Inequalities Definition

Inequality refers to an equation which is said to be greater than or less than a certain value rather than being equal to it. Quadratic inequalities are mathematical expressions involving a quadratic polynomial that are set to be greater than or less than a certain value, instead of being set equal to it.

Quadratic inequality has one of the following forms:

• ax² + bx + c > 0
• ax² + bx + c < 0
• ax² + bx + c ≥ 0
• ax² + bx + c ≤ 0

where a, b and c are constants with a ≠ 0.

How to Solve Quadratic Inequalities?

Steps to solve Quadratic Inequality include:

Step 1: Rewrite Inequality

Rewrite the inequality in standard form so that one side is zero.

Step 2: Solve Corresponding Quadratic Equation

Solve the equation ax² + bx + c = 0 to find the roots. The roots (or solutions) can be found using the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}​​

These roots divide the number line into intervals that you will test to determine the sign of the quadratic expression in each interval.

Step 3: Determine Intervals

Use the roots to divide the number line into intervals. These intervals will help determine where the quadratic expression is positive or negative.

• (−∞ , x₁​)
• (x₁ , x₂)
• (x₂ , ∞)

Step 4: Test Intervals

Pick a test point from each interval and substitute it into the quadratic expression to see if it satisfies the inequality.

Step 5: Write Solution

Based on the results of the test points, write the solution in interval notation (i.e., ≥ or ≤).

Examples to Solve Quadratic Inequalities

Example 1: Solve the quadratic inequality x² - 5x + 6 > 0.

Solution:

Step 1: Rewrite the Inequality

Inequality is already in standard form: x² - 5x + 6 > 0

Step 2: Solve the Corresponding Quadratic equation

x² - 5x + 6 = 0

(x - 2)(x - 3) = 0

Roots are x = 2 and x = 3.

Step 3: Determine Intervals

Roots divide the number line into three intervals: (-∞, 2) , (2, 3) and (3, ∞)

Step 4: Test Intervals

For (-∞, 2), pick a test point x = 0

0² -5(0) + 6 = 6 > 0 (True)

For (-∞, 2), pick a test point x = 2.5

(2.5)² -5(2.5) + 6 = -0.25 < 0 (False)

For (-∞,2), pick a test point x = 4

4² -5(4) + 6 = 2 > 0 (True)

Step 5: Write the Solution

Quadratic expression x² - 5x + 6 is greater than zero in the interval (-∞, 2) and (3, ∞)

Therefore the solution is x ∈ (-∞,2) ∪ (3,∞)

Example 2: Solve the quadratic inequality x² - 7x + 6 ≥ 0.

Solution:

Step 1: Rewrite the Inequality

Inequality is already in standard form: x² - 7x + 6 ≥ 0

Step 2: Solve the Corresponding Quadratic equation

x² - 7x + 6 = 0

(x - 2)(x - 5) = 0

Roots are x = 2 and x = 5.

Step 3: Determine Intervals

Roots divide the number line into three intervals: (-∞, 2) , (2, 5) and (5, ∞)

Step 4: Test the Intervals

For (-∞,2), pick a test point x = 0

0² -7(0) + 10 = 10 > 0 (True)

For (2,5), pick a test point x = 3

3² -7(3) + 10 = -2 < 0 (False)

For (5,∞), pick a test point x = 6

6² -7(6) + 10 = 2 > 0 (True)

Step 5: Write the Solution

Quadratic expression x² - 7x + 6 = 0. is greater than zero in the interval (-∞, 2] and [5, ∞)

Therefore the solution is x ∈ (-∞, 2] ∪ [5, ∞)

Practice Questions on Quadratic Inequalities

Questions 1. Solve the quadratic inequality x² - 4x + 3 > 0.

Questions 2. Solve the quadratic inequality x² + 2x - 8 < 0.

Questions 3. Solve the quadratic inequality x² - 3x + 2 ≤ 0.

Questions 4. Solve the quadratic inequality x² - x -12 ≥ 0.

Questions 5. Solve the quadratic inequality 2x² - 8x + 6 < 0.

Related Articles:

• Solutions of Linear Inequalities
• Quadratic Equations
• Graphical Solution of Linear Inequalities
• What is Linear Graph?