Solving Quadratic Inequality $x^2 - 5 \leq 0$ A Step-by-Step Guide

Introduction

In this comprehensive exploration, we delve into the intricacies of solving the quadratic inequality $x^2 - 5 \leq 0$. Quadratic inequalities like this one are fundamental in algebra and calculus, appearing in various contexts, from optimization problems to analyzing the behavior of functions. Understanding how to find the solution set of such inequalities is crucial for any student or professional working with mathematical models. This article aims to provide a clear, step-by-step guide to solving the given inequality, discussing the underlying principles and common pitfalls along the way. We will explore the algebraic methods for solving the inequality and visualize the solution graphically, ensuring a thorough understanding of the topic. Our main focus will be on identifying the correct solution set from the given options and justifying our answer with a rigorous mathematical approach. By the end of this discussion, you should be well-equipped to tackle similar problems and have a solid grasp of the concepts involved. This detailed exploration will not only provide the answer but also enhance your problem-solving skills in mathematics, particularly in the realm of inequalities and quadratic functions. This foundational knowledge is essential for advanced mathematical studies and real-world applications.

Understanding Quadratic Inequalities

Before we dive into the specifics of our problem, let's establish a solid understanding of quadratic inequalities in general. 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 can be written as $ax^2 + bx + c \leq 0$, $ax^2 + bx + c \geq 0$, $ax^2 + bx + c < 0$, or $ax^2 + bx + c > 0$, where a, b, and c are constants and a is not equal to zero. The solutions to these inequalities are the set of all real numbers x that satisfy the inequality. To solve a quadratic inequality, we typically follow a structured approach that involves several key steps. First, we find the roots of the corresponding quadratic equation ($ax^2 + bx + c = 0$). These roots, also known as the zeros of the quadratic function, are the points where the parabola intersects the x-axis. Next, we use these roots to divide the number line into intervals. The sign of the quadratic expression remains constant within each interval, so we can test a single point in each interval to determine whether the inequality is satisfied. Finally, we combine the intervals that satisfy the inequality to form the solution set. Understanding the behavior of quadratic functions, particularly the shape of the parabola and the location of its roots, is crucial for solving quadratic inequalities. The sign of the leading coefficient (a) determines whether the parabola opens upwards (a > 0) or downwards (a < 0), which affects the intervals where the quadratic expression is positive or negative. This foundational knowledge provides the framework for tackling specific quadratic inequalities, such as the one presented in our problem.

Solving $x^2 - 5 \leq 0$

Now, let’s tackle the specific quadratic inequality at hand: $x^2 - 5 \leq 0$. To find the solution set, we'll follow the steps outlined in the previous section. First, we need to find the roots of the corresponding quadratic equation, which is $x^2 - 5 = 0$. This equation can be solved by adding 5 to both sides, giving us $x^2 = 5$. Taking the square root of both sides, we get $x = \pm\sqrt{5}$. These roots, $-\sqrt{5}$ and $\sqrt{5}$, are critical points that divide the number line into three intervals: $(-\infty, -\sqrt{5})$, $(-\sqrt{5}, \sqrt{5})$, and $(\sqrt{5}, \infty)$. Next, we need to test a value from each interval to determine where the inequality $x^2 - 5 \leq 0$ holds true. For the interval $(-\infty, -\sqrt{5})$, let's choose $x = -3$. Plugging this into the inequality, we get $(-3)^2 - 5 = 9 - 5 = 4$, which is not less than or equal to 0. So, this interval does not satisfy the inequality. For the interval $(-\sqrt{5}, \sqrt{5})$, let's choose $x = 0$. Plugging this into the inequality, we get $(0)^2 - 5 = -5$, which is less than or equal to 0. So, this interval does satisfy the inequality. For the interval $(\sqrt{5}, \infty)$, let's choose $x = 3$. Plugging this into the inequality, we get $(3)^2 - 5 = 9 - 5 = 4$, which is not less than or equal to 0. So, this interval does not satisfy the inequality. Therefore, the solution set is the interval $[-\sqrt{5}, \sqrt{5}]$. This means that all values of x between $-\sqrt{5}$ and $\sqrt{5}$, inclusive, satisfy the inequality. This methodical approach ensures that we accurately identify the solution set by considering all possible intervals and testing representative values within them.

Identifying the Correct Option

Having solved the quadratic inequality $x^2 - 5 \leq 0$, we've determined that the solution set is the interval $[-\sqrt{5}, \sqrt{5}]$. This means we are looking for an option that represents all values of x that are greater than or equal to $-\sqrt{5}$ and less than or equal to $\sqrt{5}$. Now, let's examine the given options:

A. {x |-5 \leq x \leq \sqrt{5}} B. {x |-5 \leq x \leq 5} C. {x |-\sqrt{5} \leq x \leq 5} D. {x |-\sqrt{5} \leq x \leq \sqrt{5}}

By comparing our solution set $[-\sqrt{5}, \sqrt{5}]$ with the provided options, it becomes clear that option D, {x |-\sqrt{5} \leq x \leq \sqrt{5}}, perfectly matches our derived solution. This option correctly captures the interval of x values that satisfy the inequality. The other options contain incorrect bounds or ranges for x. Option A includes values less than $-\sqrt{5}$, option B has incorrect bounds altogether, and option C includes values greater than $\sqrt{5}$. Therefore, by carefully analyzing the solution set and comparing it to the given options, we can confidently identify the correct answer. This process underscores the importance of both accurately solving the inequality and meticulously matching the solution to the available choices. The correct identification of the solution set requires a precise understanding of interval notation and the implications of the inequality's bounds.

Conclusion

In conclusion, we have successfully solved the quadratic inequality $x^2 - 5 \leq 0$ and identified the correct solution set. Our step-by-step approach involved finding the roots of the corresponding quadratic equation, dividing the number line into intervals, testing values within each interval, and then comparing our solution with the given options. We determined that the solution set is $[-\sqrt{5}, \sqrt{5}]$, which corresponds to option D: {x |$-\sqrt{5} \leq x \leq \sqrt{5}$}. This exercise highlights the importance of a systematic approach to solving quadratic inequalities and the necessity of understanding the underlying principles of quadratic functions and inequalities. By thoroughly understanding the steps involved, from finding the roots to testing intervals, we can confidently tackle similar problems. The ability to accurately solve quadratic inequalities is a valuable skill in mathematics, with applications in various fields, including physics, engineering, and economics. Moreover, this process reinforces critical thinking and problem-solving skills, which are essential for academic and professional success. By mastering these techniques, students and professionals can confidently approach complex mathematical problems and arrive at accurate solutions. The methodical approach presented here provides a solid foundation for further exploration of mathematical concepts and their real-world applications.