Solving 4x² - 36 = 0 Finding The Solution Set

In the realm of mathematics, solving equations is a fundamental skill. Quadratic equations, with their characteristic $x^2$ term, appear frequently in various mathematical and scientific contexts. One such equation is $4x^2 - 36 = 0$. This article delves into the process of finding the solution set for this equation, providing a step-by-step explanation that is accessible to both students and enthusiasts. We will explore the algebraic techniques involved, discuss the nature of the solutions, and highlight the significance of understanding solution sets in the broader context of mathematics. Mastering the solution of quadratic equations like this one lays a crucial foundation for tackling more complex mathematical problems. Understanding the solution set not only provides the numerical answers but also offers insights into the behavior of the equation and its graphical representation. So, let's embark on this journey of mathematical exploration and unravel the mysteries of the solution set for the equation $4x^2 - 36 = 0$.

Understanding Quadratic Equations

Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (typically denoted as x) is 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants, and a is not equal to 0. The solutions to a quadratic equation are also known as its roots or zeros. These roots represent the values of x that satisfy the equation, making the expression equal to zero. Finding these roots is a core concept in algebra and has wide-ranging applications in various fields, including physics, engineering, and economics. The solutions can be real numbers or complex numbers, depending on the specific coefficients of the equation. In the context of graphs, the real roots of a quadratic equation correspond to the x-intercepts of the parabola represented by the equation. Understanding the nature and behavior of quadratic equations is essential for solving problems involving parabolic trajectories, optimization problems, and many other real-world scenarios. The solution set, which is the set of all roots, provides a complete picture of the values that satisfy the equation. By mastering the techniques for solving quadratic equations, one can gain a deeper appreciation for the elegance and power of algebraic methods in problem-solving.

Solving 4x² - 36 = 0: A Step-by-Step Approach

To determine the solution set of the quadratic equation $4x^2 - 36 = 0$, we will employ a methodical, step-by-step approach. This ensures clarity and accuracy in our solution. Our primary goal is to isolate the variable x and identify the values that make the equation true. The following steps outline our strategy:

1. Isolate the x² term: Our initial step involves isolating the $x^2$ term on one side of the equation. This can be achieved by adding 36 to both sides of the equation. This operation maintains the equality and moves us closer to isolating x. The resulting equation is $4x^2 = 36$.
2. Divide to isolate x²: Next, we divide both sides of the equation by 4. This isolates the $x^2$ term completely, giving us $x^2 = 9$. This step simplifies the equation and brings us closer to finding the value(s) of x.
3. Take the square root: To solve for x, we take the square root of both sides of the equation. It's crucial to remember that the square root operation yields both positive and negative solutions. This is because both the positive and negative square roots, when squared, will result in the original number. Therefore, we have $x = ±√9$.
4. Determine the solutions: Finally, we simplify the square root of 9, which is 3. This gives us two solutions: $x = 3$ and $x = -3$. These are the two values that, when substituted into the original equation, will make the equation true.
5. Express the solution set: The solution set is the collection of all values of x that satisfy the equation. In this case, the solution set is {-3, 3}. This set represents the complete set of solutions to the quadratic equation $4x^2 - 36 = 0$.

Alternative Method: Factoring

Factoring is another powerful technique for solving quadratic equations. It involves expressing the quadratic expression as a product of two linear expressions. This method can be particularly efficient when the quadratic equation can be easily factored. Let's apply factoring to the equation $4x^2 - 36 = 0$.

1. Recognize the difference of squares: Notice that the equation can be rewritten as $(2x)^2 - 6^2 = 0$. This is in the form of a difference of squares, which has a well-known factoring pattern: $a^2 - b^2 = (a + b)(a - b)$.
2. Factor the equation: Applying the difference of squares pattern, we can factor the equation as $(2x + 6)(2x - 6) = 0$. This step transforms the quadratic equation into a product of two linear factors.
3. Set each factor to zero: For the product of two factors to be zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero: $2x + 6 = 0$ and $2x - 6 = 0$.
4. Solve for x: Solving the first equation, $2x + 6 = 0$, we subtract 6 from both sides to get $2x = -6$, and then divide by 2 to get $x = -3$. Solving the second equation, $2x - 6 = 0$, we add 6 to both sides to get $2x = 6$, and then divide by 2 to get $x = 3$.
5. Verify the solutions: We have obtained the solutions $x = -3$ and $x = 3$, which are the same solutions we found using the previous method. This confirms the accuracy of both methods.
6. State the solution set: The solution set is {-3, 3}, which is consistent with our earlier result. This demonstrates that factoring is a viable alternative method for solving quadratic equations.

The factoring method highlights the importance of recognizing patterns in algebraic expressions. By identifying the difference of squares, we were able to efficiently factor the equation and arrive at the solutions. This technique is a valuable tool in the mathematician's toolkit and can often simplify the process of solving quadratic equations.

The Significance of the Solution Set

The solution set of an equation, including quadratic equations, is more than just a collection of numbers; it represents the complete set of values that satisfy the equation. Understanding the significance of the solution set provides valuable insights into the nature of the equation and its implications. In the case of $4x^2 - 36 = 0$, the solution set {-3, 3} tells us that only two values of x, -3 and 3, will make the equation true. Any other value substituted for x will result in an inequality.

The solution set also has a graphical interpretation. The equation $4x^2 - 36 = 0$ can be viewed as a parabola when graphed. The solutions, -3 and 3, correspond to the x-intercepts of the parabola, which are the points where the parabola intersects the x-axis. This connection between the algebraic solutions and the graphical representation is a fundamental concept in mathematics.

Furthermore, the solution set provides information about the number of real roots the equation has. In this case, the equation has two distinct real roots. Quadratic equations can also have one real root (a repeated root) or no real roots (two complex roots). The nature of the roots is determined by the discriminant of the quadratic equation, which is a value derived from the coefficients of the equation.

In practical applications, the solution set can represent real-world quantities. For example, if the equation models the trajectory of a projectile, the solutions might represent the times at which the projectile reaches a certain height. Understanding the solution set in these contexts allows us to make predictions and solve problems in various fields, such as physics, engineering, and finance.

In conclusion, the solution set is a crucial concept in mathematics that provides a comprehensive understanding of the solutions to an equation. It connects algebraic solutions to graphical representations and has practical applications in numerous fields. By mastering the techniques for finding solution sets, we gain a powerful tool for solving mathematical problems and interpreting their real-world implications.

Conclusion

In this exploration, we have successfully determined the solution set of the quadratic equation $4x^2 - 36 = 0$. Through a step-by-step algebraic approach and the alternative method of factoring, we found that the solution set is {-3, 3}. These two values, -3 and 3, are the only values of x that satisfy the equation, making it a true statement. This process highlights the importance of understanding quadratic equations and the various techniques available for solving them.

We discussed the general form of quadratic equations and emphasized the significance of isolating the variable to find the solutions. We also explored the method of factoring, demonstrating its efficiency in cases where the quadratic expression can be easily factored. The connection between the algebraic solutions and the graphical representation of the equation was also discussed, highlighting the fact that the solutions correspond to the x-intercepts of the parabola.

Furthermore, we underscored the importance of the solution set as a comprehensive representation of the solutions to an equation. It provides valuable insights into the nature of the equation and its implications in various contexts. Understanding the solution set allows us to not only solve mathematical problems but also interpret their real-world significance.

The ability to solve quadratic equations is a fundamental skill in mathematics with applications in numerous fields. By mastering the techniques discussed in this article, you will be well-equipped to tackle a wide range of mathematical problems and gain a deeper appreciation for the elegance and power of algebraic methods. The solution set is a key concept that provides a complete understanding of the solutions to an equation, making it an essential tool in the mathematician's toolkit. As you continue your mathematical journey, remember that the skills and concepts learned here will serve as a strong foundation for more advanced topics and applications.