In the realm of mathematics, quadratic inequalities play a crucial role in understanding the behavior of quadratic functions and their solutions. These inequalities, which involve a quadratic expression compared to a constant, zero, or another expression, often appear in various mathematical contexts, including calculus, optimization problems, and real-world applications. This article delves into a step-by-step approach to solving the quadratic inequality x² + 4x - 21 ≥ 0, providing a clear understanding of the underlying concepts and techniques involved. By the end of this guide, you'll be equipped with the knowledge and skills necessary to tackle similar quadratic inequalities with confidence.
Understanding Quadratic Inequalities
Before diving into the specific problem, it's essential to grasp the fundamental concepts of quadratic inequalities. A quadratic inequality is an inequality that involves a quadratic expression, which is a polynomial of degree two. 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 solutions to a quadratic inequality represent the set of real numbers that satisfy the given inequality. Graphically, these solutions correspond to the intervals on the x-axis where the parabola represented by the quadratic expression lies above or below the x-axis, depending on the inequality sign.
To effectively solve quadratic inequalities, a multi-faceted approach is required, encompassing algebraic manipulation, graphical interpretation, and logical reasoning. The initial step typically involves transforming the inequality into a standard form, followed by the identification of critical points, which serve as pivotal boundaries in determining the solution intervals. Subsequently, these critical points are employed to partition the number line into distinct intervals, and the sign of the quadratic expression is assessed within each interval to ascertain whether it satisfies the given inequality. A clear understanding of these concepts is foundational for successfully navigating the intricacies of solving quadratic inequalities.
Step-by-Step Solution: x² + 4x - 21 ≥ 0
Let's embark on a step-by-step journey to solve the quadratic inequality x² + 4x - 21 ≥ 0. This process will not only provide the solution to this specific problem but also equip you with a general methodology applicable to a wide range of quadratic inequalities. By meticulously following each step, you'll gain a profound understanding of the underlying principles and techniques, enabling you to confidently tackle similar mathematical challenges.
Step 1: Factor the Quadratic Expression
The first crucial step is to factor the quadratic expression x² + 4x - 21. Factoring allows us to rewrite the expression as a product of two linear factors, which will be instrumental in identifying the critical points. To factor the quadratic, we seek two numbers that multiply to -21 (the constant term) and add up to 4 (the coefficient of the x term). These numbers are 7 and -3. Therefore, we can factor the expression as follows:
x² + 4x - 21 = (x + 7)(x - 3)
This factorization transforms the quadratic inequality into:
(x + 7)(x - 3) ≥ 0
Factoring the quadratic expression is a fundamental step in solving quadratic inequalities. It allows us to identify the critical points, which are the values of x that make the expression equal to zero. These critical points serve as boundaries for the intervals where the expression is either positive or negative. By factoring the quadratic, we've taken the first step towards unraveling the solution set of the inequality.
Step 2: Find the Critical Points
The critical points are the values of x that make the quadratic expression equal to zero. These points are crucial because they divide the number line into intervals where the expression is either positive or negative. To find the critical points, we set each factor equal to zero and solve for x:
- x + 7 = 0 => x = -7
- x - 3 = 0 => x = 3
Thus, the critical points are x = -7 and x = 3. These points are the roots of the quadratic equation x² + 4x - 21 = 0. They represent the x-intercepts of the parabola defined by the quadratic expression. The critical points are the linchpins in determining the solution intervals of the inequality, as they mark the transitions between positive and negative values of the quadratic expression.
Step 3: Create a Sign Chart
A sign chart is a powerful tool for visualizing the sign of the quadratic expression in different intervals. We create a sign chart by drawing a number line and marking the critical points on it. The critical points divide the number line into intervals. In each interval, the quadratic expression will have a constant sign (either positive or negative). To determine the sign in each interval, we choose a test value within the interval and evaluate the expression at that value. The sign of the result will be the sign of the expression in that interval.
Let's create a sign chart for (x + 7)(x - 3) ≥ 0:
- Draw a number line and mark the critical points -7 and 3.
- This divides the number line into three intervals: (-∞, -7), (-7, 3), and (3, ∞).
- Choose a test value in each interval:
- Interval (-∞, -7): Let x = -8. (-8 + 7)(-8 - 3) = (-1)(-11) = 11 > 0
- Interval (-7, 3): Let x = 0. (0 + 7)(0 - 3) = (7)(-3) = -21 < 0
- Interval (3, ∞): Let x = 4. (4 + 7)(4 - 3) = (11)(1) = 11 > 0
The sign chart will look like this:
Interval | (-∞, -7) | (-7, 3) | (3, ∞)
------------|----------|---------|--------
(x + 7) | - | + | +
(x - 3) | - | - | +
(x + 7)(x - 3) | + | - | +
The sign chart is a visual representation of the behavior of the quadratic expression across different intervals. It allows us to quickly identify the intervals where the expression is positive, negative, or zero. By analyzing the sign chart, we can determine the solution set of the inequality.
Step 4: Determine the Solution Set
We are looking for the intervals where (x + 7)(x - 3) ≥ 0. This means we want the intervals where the expression is either positive or equal to zero. From the sign chart, we see that the expression is positive in the intervals (-∞, -7) and (3, ∞). It is equal to zero at the critical points x = -7 and x = 3. Therefore, the solution set includes these critical points.
Thus, the solution to the inequality x² + 4x - 21 ≥ 0 is:
x ≤ -7 or x ≥ 3
In interval notation, the solution set is:
(-∞, -7] ∪ [3, ∞)
The solution set represents all the real numbers that satisfy the inequality. In this case, it includes all numbers less than or equal to -7 and all numbers greater than or equal to 3. This means that any value of x within these intervals, when substituted into the original inequality, will make the inequality true.
Visualizing the Solution
To further solidify our understanding, let's visualize the solution graphically. The quadratic expression x² + 4x - 21 represents a parabola that opens upwards (since the coefficient of x² is positive). The critical points, x = -7 and x = 3, are the x-intercepts of the parabola. The inequality x² + 4x - 21 ≥ 0 asks for the values of x where the parabola is above or on the x-axis.
By sketching the parabola, we can see that it is above the x-axis for x < -7 and x > 3, and it is on the x-axis at x = -7 and x = 3. This graphical representation confirms our algebraic solution, providing a visual understanding of the solution set.
Key Takeaways and General Approach
Solving quadratic inequalities involves a systematic approach that combines algebraic manipulation, graphical interpretation, and logical reasoning. Here's a recap of the key steps and a general approach to solving quadratic inequalities:
- Rewrite the inequality: If necessary, rewrite the inequality so that one side is zero.
- Factor the quadratic expression: Factor the quadratic expression into two linear factors.
- Find the critical points: Set each factor equal to zero and solve for x. These are the critical points.
- Create a sign chart: Draw a number line and mark the critical points. Choose a test value in each interval and determine the sign of the expression in that interval.
- Determine the solution set: Identify the intervals where the expression satisfies the inequality. Include the critical points if the inequality is non-strict (≥ or ≤).
- Express the solution: Write the solution set in interval notation or inequality notation.
By mastering this approach, you'll be well-equipped to solve a wide range of quadratic inequalities. Remember to practice regularly and apply these techniques to various problems to enhance your understanding and problem-solving skills.
Conclusion
Solving the quadratic inequality x² + 4x - 21 ≥ 0 has been a journey through the core concepts of quadratic inequalities. We've explored the importance of factoring, identifying critical points, creating sign charts, and interpreting solutions both algebraically and graphically. This comprehensive guide has provided a framework for tackling similar problems, empowering you to confidently navigate the world of quadratic inequalities. Remember, the key to success lies in practice and a thorough understanding of the underlying principles. So, keep exploring, keep solving, and keep mastering the art of mathematics!