Quadratic Function Inequalities

The vertex of the parabola is the point where the function reaches its minimum or maximum value. The x-coordinate of the vertex can be found using the formula x = -b / (2a). The y-coordinate of the vertex can be found by substituting this x-value back into the quadratic function.

Types of Quadratic Function Inequalities

Quadratic function inequalities can be categorized into three types:

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

Each type requires a different approach to solve, but the general steps remain the same.

Solving Quadratic Function Inequalities

To solve a quadratic function inequality, follow these steps:

1. Rewrite the inequality in standard form: Ensure the inequality is in the form ax² + bx + c < 0, ax² + bx + c > 0, ax² + bx + c ≤ 0, or ax² + bx + c ≥ 0.
2. Find the roots of the corresponding quadratic equation: Solve ax² + bx + c = 0 using the quadratic formula x = [-b ± √(b² - 4ac)] / (2a).
3. Determine the intervals: The roots divide the number line into intervals. Test a point in each interval to determine where the inequality holds true.
4. Consider the parabola's orientation: If a > 0, the parabola opens upwards, and the inequality ax² + bx + c < 0 holds between the roots. If a < 0, the parabola opens downwards, and the inequality ax² + bx + c > 0 holds between the roots.

💡 Note: When dealing with inequalities of the form ax² + bx + c ≤ 0 or ax² + bx + c ≥ 0, include the roots in the solution set if the inequality is non-strict.

Practical Examples

Let's solve a few examples to illustrate the process.

Example 1: x² - 3x + 2 < 0

1. Rewrite the inequality: The inequality is already in standard form.

2. Find the roots: Solve x² - 3x + 2 = 0. The roots are x = 1 and x = 2.

3. Determine the intervals: The roots divide the number line into three intervals: (-∞, 1), (1, 2), and (2, ∞).

4. Test the intervals: Test a point in each interval. For example, x = 0 in (-∞, 1) gives 0² - 3(0) + 2 = 2 > 0. x = 1.5 in (1, 2) gives 1.5² - 3(1.5) + 2 = -0.25 < 0. x = 3 in (2, ∞) gives 3² - 3(3) + 2 = 2 > 0.

5. Conclusion: The inequality x² - 3x + 2 < 0 holds for 1 < x < 2.

Example 2: -x² + 4x - 3 ≥ 0

1. Rewrite the inequality: The inequality is already in standard form.

2. Find the roots: Solve -x² + 4x - 3 = 0. The roots are x = 1 and x = 3.

3. Determine the intervals: The roots divide the number line into three intervals: (-∞, 1), (1, 3), and (3, ∞).

4. Test the intervals: Test a point in each interval. For example, x = 0 in (-∞, 1) gives -0² + 4(0) - 3 = -3 < 0. x = 2 in (1, 3) gives -2² + 4(2) - 3 = 1 ≥ 0. x = 4 in (3, ∞) gives -4² + 4(4) - 3 = -3 < 0.

5. Conclusion: The inequality -x² + 4x - 3 ≥ 0 holds for 1 ≤ x ≤ 3.

Graphical Representation

Graphing the quadratic function can provide a visual representation of the solution set. The x-intercepts of the graph correspond to the roots of the quadratic equation, and the intervals where the graph is above or below the x-axis indicate where the inequality holds true.

For example, consider the inequality x² - 3x + 2 < 0. The graph of y = x² - 3x + 2 is a parabola opening upwards with roots at x = 1 and x = 2. The parabola is below the x-axis between these roots, confirming that the inequality holds for 1 < x < 2.

Special Cases

There are a few special cases to consider when solving Quadratic Function Inequalities:

• No real roots: If the discriminant b² - 4ac < 0, the quadratic equation has no real roots. The parabola does not intersect the x-axis, and the inequality ax² + bx + c < 0 or ax² + bx + c > 0 holds for all x depending on the sign of a.
• One real root: If the discriminant b² - 4ac = 0, the quadratic equation has one real root. The parabola touches the x-axis at this point, and the inequality ax² + bx + c ≤ 0 or ax² + bx + c ≥ 0 holds for all x depending on the sign of a.
• Complex roots: If the discriminant b² - 4ac < 0, the quadratic equation has complex roots. The parabola does not intersect the x-axis, and the inequality ax² + bx + c < 0 or ax² + bx + c > 0 holds for all x depending on the sign of a.

💡 Note: When the discriminant is zero, the parabola touches the x-axis at exactly one point, and the inequality will hold for all x if it includes equality.

Applications of Quadratic Function Inequalities

Quadratic function inequalities have numerous applications in various fields. Here are a few examples:

• Physics: Quadratic inequalities can model the motion of objects under gravity, where the height of an object is a quadratic function of time.
• Engineering: In structural engineering, quadratic inequalities can be used to determine the safe load limits for beams and other structures.
• Economics: Quadratic functions can model profit or cost functions, and inequalities can be used to determine the range of production levels that result in a profit.

Summary of Key Points

Solving Quadratic Function Inequalities involves understanding the properties of quadratic functions, finding the roots of the corresponding quadratic equation, and determining the intervals where the inequality holds true. Graphing the quadratic function can provide a visual representation of the solution set. Special cases, such as no real roots or complex roots, require careful consideration. Quadratic function inequalities have wide-ranging applications in physics, engineering, economics, and other fields.