Write a quadratic inequality represented by the graph. - brainly.com

Types of Quadratic Function Inequalities

Quadratic function inequalities can be categorized into three main types:

• ax² + bx + c > 0
• ax² + bx + c < 0
• ax² + bx + c &geq 0
• ax² + bx + c &leq 0

Each type requires a specific approach to solve, but the general method involves finding the roots of the quadratic equation and analyzing the intervals between these roots.

Solving Quadratic Function Inequalities

To solve a quadratic function inequality, follow these steps:

1. Find the roots of the quadratic equation: Set the quadratic expression equal to zero and solve for x. This can be done using the quadratic formula, x = [-b ± √(b² - 4ac)] / (2a).
2. Determine the intervals: The roots divide the number line into intervals. Test a point in each interval to determine if it satisfies the inequality.
3. Analyze the parabola: Consider the direction in which the parabola opens (upwards or downwards) to determine the solution set.

Step-by-Step Example

Let’s solve the inequality x² - 3x + 2 < 0 step by step.

1. Find the roots: Set the quadratic expression equal to zero: x² - 3x + 2 = 0. Factor the quadratic to get (x - 1)(x - 2) = 0. The roots are x = 1 and x = 2.
2. Determine the intervals: The roots divide the number line into three intervals: (-∞, 1), (1, 2), and (2, ∞).
3. Test the intervals: Choose a test point in each interval and substitute it into the inequality.
   • For (-∞, 1), choose x = 0: 0² - 3(0) + 2 = 2 > 0 (does not satisfy the inequality).
   • For (1, 2), choose x = 1.5: 1.5² - 3(1.5) + 2 = -0.25 < 0 (satisfies the inequality).
   • For (2, ∞), choose x = 3: 3² - 3(3) + 2 = 2 > 0 (does not satisfy the inequality).
4. Analyze the parabola: The parabola opens upwards (since the coefficient of x² is positive). Therefore, the solution set is the interval where the parabola is below the x-axis, which is (1, 2).

💡 Note: The solution set for x² - 3x + 2 < 0 is (1, 2). For the inequality x² - 3x + 2 &geq 0, the solution set would include the roots, making it (-∞, 1] ∪ [2, ∞).

Special Cases

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

• No real roots: If the discriminant (b² - 4ac) is negative, the quadratic has no real roots. The inequality ax² + bx + c > 0 is satisfied for all x if a > 0, and ax² + bx + c < 0 is satisfied for all x if a < 0.
• Repeated roots: If the discriminant is zero, the quadratic has a repeated root. The inequality ax² + bx + c > 0 is satisfied for all x the root if a > 0, and ax² + bx + c < 0 is satisfied for all x the root if a < 0.

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. The intervals where the graph is above or below the x-axis indicate where the quadratic expression is positive or negative, respectively.

Practical Applications

Quadratic Function Inequalities have numerous practical applications. Here are a few examples:

• Physics: Modeling the motion of objects under gravity, where the height of an object is a quadratic function of time.
• Engineering: Designing structures that must withstand certain loads, where the stress on a material is a quadratic function of the applied force.
• Economics: Analyzing profit and loss, where the profit function is often quadratic in terms of the number of units produced.

Solving Systems of Quadratic Inequalities

Sometimes, you may encounter systems of quadratic inequalities that need to be solved simultaneously. The process involves finding the solution sets for each inequality and then determining their intersection.

For example, consider the system:

x² - 3x + 2 < 0 x² - 5x + 6 &geq 0

First, solve each inequality separately:

• x² - 3x + 2 < 0 has the solution set (1, 2).
• x² - 5x + 6 &geq 0 has the solution set (-∞, 2] ∪ [3, ∞).

The intersection of these solution sets is (1, 2].

Common Mistakes to Avoid

When solving Quadratic Function Inequalities, it’s important to avoid common mistakes:

• Forgetting to test the intervals: Always test a point in each interval to determine if it satisfies the inequality.
• Ignoring the direction of the parabola: The direction in which the parabola opens (upwards or downwards) affects the solution set.
• Not considering the equality case: For inequalities of the form &geq or &leq, include the roots in the solution set if they satisfy the inequality.

By keeping these points in mind, you can avoid common pitfalls and solve quadratic function inequalities more accurately.

In conclusion, mastering Quadratic Function Inequalities involves understanding the basics of quadratic functions, knowing the steps to solve the inequalities, and practicing with various examples. Whether you’re a student preparing for an exam or a professional applying these concepts in your field, a solid grasp of quadratic function inequalities is invaluable. With practice and attention to detail, you can become proficient in solving these inequalities and applying them to real-world problems.