Solving Quadratic Inequality F(x) = X^2 + 4x - 21 ≥ 0
This article provides a step-by-step guide to solving the inequality f(x) = x² + 4x - 21 ≥ 0. We will explore the process of finding the roots of the quadratic equation, analyzing the intervals defined by these roots, and determining the solution set for the inequality. This comprehensive approach ensures a clear understanding of the solution and the underlying concepts.
Understanding Quadratic Inequalities
Quadratic inequalities are mathematical expressions that compare a quadratic expression to a value, often zero. Solving these inequalities involves finding the range(s) of x values that satisfy the given condition. The general form of a quadratic inequality is ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, or ax² + bx + c ≤ 0, where a, b, and c are constants and a ≠ 0. The key to solving these inequalities lies in understanding the behavior of quadratic functions and their graphical representation as parabolas.
Key Concepts
- Quadratic Function: A function of the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola.
- Roots/Zeros: The values of x for which f(x) = 0. These are the points where the parabola intersects the x-axis. The roots can be found by factoring the quadratic equation, using the quadratic formula, or completing the square.
- Parabola: The U-shaped curve that represents the graph of a quadratic function. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards.
- Intervals: The x-axis is divided into intervals by the roots of the quadratic equation. Within each interval, the quadratic function is either strictly positive or strictly negative.
Step-by-Step Solution for f(x) = x² + 4x - 21 ≥ 0
To solve the inequality f(x) = x² + 4x - 21 ≥ 0, we will follow a structured approach:
- Find the roots of the quadratic equation: Set f(x) = 0 and solve for x.
- Determine the intervals: The roots divide the x-axis into intervals. These intervals are the regions where the quadratic function's value has a consistent sign (either positive or negative).
- Test each interval: Choose a test value within each interval and evaluate f(x). This will reveal whether the function is positive or negative in that interval.
- Identify the solution set: Based on the sign of f(x) in each interval, determine the intervals that satisfy the inequality f(x) ≥ 0.
1. Finding the Roots of the Quadratic Equation
The first step in solving the inequality x² + 4x - 21 ≥ 0 is to find the roots of the corresponding quadratic equation:
x² + 4x - 21 = 0
We can solve this equation by factoring. We are looking for two numbers that multiply to -21 and add to 4. These numbers are 7 and -3. Thus, we can factor the quadratic as:
(x + 7)(x - 3) = 0
Setting each factor equal to zero gives us the roots:
x + 7 = 0 => x = -7
x - 3 = 0 => x = 3
Therefore, the roots of the quadratic equation are x = -7 and x = 3. These roots are crucial because they divide the number line into intervals where the function's sign remains constant.
2. Determining the Intervals
The roots x = -7 and x = 3 divide the number line into three intervals:
- (-∞, -7)
- (-7, 3)
- (3, ∞)
These intervals are the regions we will test to determine where the inequality x² + 4x - 21 ≥ 0 is satisfied. The behavior of the quadratic function within each interval is consistent, meaning the function will be either entirely positive or entirely negative within a given interval.
3. Testing Each Interval
To determine the sign of f(x) = x² + 4x - 21 in each interval, we will choose a test value within each interval and evaluate the function:
-
Interval (-∞, -7):
Choose a test value, for instance, x = -8:
f(-8) = (-8)² + 4(-8) - 21 = 64 - 32 - 21 = 11
Since f(-8) = 11 > 0, the function is positive in this interval.
-
Interval (-7, 3):
Choose a test value, for instance, x = 0:
f(0) = (0)² + 4(0) - 21 = -21
Since f(0) = -21 < 0, the function is negative in this interval.
-
Interval (3, ∞):
Choose a test value, for instance, x = 4:
f(4) = (4)² + 4(4) - 21 = 16 + 16 - 21 = 11
Since f(4) = 11 > 0, the function is positive in this interval.
By testing these values, we have a clear picture of the sign of the function in each interval. This information is essential for identifying the solution set of the inequality.
4. Identifying the Solution Set
We are looking for the intervals where f(x) = x² + 4x - 21 ≥ 0. From our tests, we found that:
- f(x) > 0 in the interval (-∞, -7)
- f(x) < 0 in the interval (-7, 3)
- f(x) > 0 in the interval (3, ∞)
Since we need f(x) ≥ 0, we include the intervals where f(x) > 0 and the points where f(x) = 0, which are the roots x = -7 and x = 3. Thus, the solution set includes the intervals (-∞, -7] and [3, ∞). The square brackets indicate that the endpoints are included in the solution.
Therefore, the solution to the inequality x² + 4x - 21 ≥ 0 is:
x ∈ (-∞, -7] ∪ [3, ∞)
Final Answer
The solution to the inequality f(x) = x² + 4x - 21 ≥ 0 is:
The solution is {(-∞, -7] ∪ [3, ∞)}.
This result indicates that the function f(x) is greater than or equal to zero for all x values less than or equal to -7 and for all x values greater than or equal to 3. This comprehensive solution provides a clear understanding of the behavior of the quadratic function and its relationship to the given inequality.
Conclusion
Solving quadratic inequalities involves a systematic approach that includes finding the roots, identifying intervals, testing values within those intervals, and determining the solution set. By following these steps, we can effectively solve inequalities such as f(x) = x² + 4x - 21 ≥ 0. Understanding these concepts is crucial for various applications in mathematics and other fields.
By breaking down the problem into manageable steps and providing clear explanations, this article aims to enhance the understanding of quadratic inequalities and their solutions. The detailed walkthrough ensures that readers can confidently tackle similar problems in the future.
