Solving quadratic inequalities

Solving Quadratic Inequalities

Quadratic inequalities are inequalities that involve a quadratic expression. A quadratic expression is a polynomial of degree two, which can be written in the standard form:

ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c >= 0, or ax² + bx + c <= 0

where a, b, and c are real numbers and a ≠ 0. To solve these inequalities, you will follow several steps.

Step 1: Solve the Corresponding Quadratic Equation

The first step in solving a quadratic inequality is to solve the corresponding quadratic equation. This means setting the inequality to zero and finding the roots (solutions) of the equation:

ax² + bx + c = 0

You can use the quadratic formula to find the roots:

x = (-b ± √(b² - 4ac)) / (2a)

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

1. First, find the roots of the equation x² - 5x + 6 = 0.

x = (5 ± √(5² - 4 × 1 × 6)) / (2 × 1)

2. Calculate the discriminant (b² - 4ac):

5² - 4 × 1 × 6 = 25 - 24 = 1

3. Since the discriminant is positive, there are two distinct real roots:

x = (5 ± √1) / 2

4. Calculate the roots:

x = (5 + 1) / 2 = 3

x = (5 - 1) / 2 = 2

The roots are x = 2 and x = 3.

Step 2: Determine the Intervals

Next, you need to determine the intervals created by the roots. The roots divide the number line into intervals. For the roots x = 2 and x = 3, the intervals are:

• (-∞, 2)
• (2, 3)
• (3, ∞)

Step 3: Test the Intervals

Choose a test point from each interval to determine whether the inequality holds true for that interval. Substitute the test point into the original inequality.

Example: Test the intervals for the inequality x² - 5x + 6 > 0.

1. Test point in (-∞, 2): Choose x = 0.

0² - 5(0) + 6 = 6 > 0 (True)

2. Test point in (2, 3): Choose x = 2.5.

(2.5)² - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25 > 0 (False)

3. Test point in (3, ∞): Choose x = 4.

4² - 5(4) + 6 = 16 - 20 + 6 = 2 > 0 (True)

Step 4: Write the Solution

Based on the test results, the solution to the inequality x² - 5x + 6 > 0 is:

x ∈ (-∞, 2) ∪ (3, ∞)

This means that the inequality holds true for values of x less than 2 and greater than 3.

Step 5: Consider Equality

If the inequality is non-strict (≥ or ≤), you must also consider the points where the quadratic expression equals zero (the roots). For instance, if the inequality is x² - 5x + 6 ≥ 0, the solution would include the roots:

x ∈ (-∞, 2] ∪ [3, ∞)

Common Mistakes

Graphical Representation of Quadratic Inequalities

Graphing the quadratic function can help visualise the solution to the inequality. The graph of a quadratic function is a parabola. The roots of the equation are where the parabola intersects the x-axis. The regions where the parabola is above (for > or ≥) or below (for < or ≤) the x-axis represent the solution to the inequality.

Example: Graph y = x² - 5x + 6.

The roots are x = 2 and x = 3. The parabola opens upwards since the coefficient of x² is positive (a = 1). The solution to the inequality x² - 5x + 6 > 0 corresponds to the regions where the graph is above the x-axis.

Summary

• To solve quadratic inequalities, first solve the corresponding quadratic equation.
• Determine the intervals created by the roots.
• Test each interval with a test point.
• Write the solution based on the test results.
• Include the roots in the solution if the inequality is non-strict.

Check your understanding

1. What are the steps to solve the quadratic inequality x² + 4x + 3 < 0?
2. How do you determine the intervals after finding the roots?
3. What is the significance of the test points in each interval?
4. What would the solution be for the inequality x² - 2x - 8 ≥ 0?