Solving quadratic equations is a fundamental concept in algebra, and understanding the various methods to find solutions is crucial for students and anyone working with mathematical models. This article dives deep into the quadratic equation x² - 25 = 0, exploring different approaches to arrive at the correct solutions and providing a comprehensive explanation for each step. Let's embark on this mathematical journey to find the solutions and solidify our understanding of quadratic equations.
Understanding Quadratic Equations
Before we tackle the specific equation, let's establish a strong foundation by defining what a quadratic equation is and reviewing its standard form. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is expressed as ax² + bx + c = 0, where a, b, and c are constants, and 'a' is not equal to zero. The solutions to a quadratic equation are also known as roots or zeros, which represent the values of 'x' that satisfy the equation. These solutions correspond to the points where the parabola represented by the quadratic equation intersects the x-axis on a graph. Quadratic equations arise in various real-world applications, such as physics, engineering, and economics, making their understanding paramount. Consider projectile motion, where the path of an object through the air can be modeled using a quadratic equation. Similarly, in engineering, quadratic equations are used in the design of bridges and other structures. In economics, quadratic functions can model cost, revenue, and profit curves. Mastering the techniques for solving quadratic equations equips us with powerful tools for analyzing and solving problems in diverse fields. One common misconception is to think that all quadratic equations have two distinct real solutions, but this is not always the case. Some quadratic equations may have two distinct real solutions, one repeated real solution, or two complex solutions. The nature of the solutions depends on the discriminant, which we will touch upon later. The process of solving a quadratic equation involves isolating the variable 'x' and finding the values that make the equation true. There are several methods to accomplish this, including factoring, completing the square, and using the quadratic formula. Each method has its own advantages and is suitable for different types of quadratic equations. By grasping the underlying principles of quadratic equations and the various methods for solving them, we can confidently approach a wide range of mathematical problems and real-world applications. In the following sections, we will delve into the specific equation x² - 25 = 0 and explore different solution methods in detail.
Method 1: Factoring
One of the most efficient methods to solve quadratic equations, when applicable, is factoring. Factoring involves expressing the quadratic equation as a product of two linear factors. This method relies on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Applying this to our equation, x² - 25 = 0, we can recognize it as a difference of squares. The difference of squares pattern is a fundamental algebraic identity that can be expressed as a² - b² = (a + b)(a - b). In our case, x² corresponds to a², and 25 corresponds to b², where b = 5. Therefore, we can rewrite the equation as x² - 5² = 0. Applying the difference of squares pattern, we factor the left side of the equation as (x + 5)(x - 5) = 0. Now that we have factored the quadratic equation, we can apply the zero-product property. This property states that if (x + 5)(x - 5) = 0, then either (x + 5) = 0 or (x - 5) = 0. Setting each factor equal to zero gives us two linear equations: x + 5 = 0 and x - 5 = 0. Solving the first equation, x + 5 = 0, we subtract 5 from both sides to isolate 'x', resulting in x = -5. Solving the second equation, x - 5 = 0, we add 5 to both sides to isolate 'x', resulting in x = 5. Thus, the solutions to the quadratic equation x² - 25 = 0 are x = 5 and x = -5. Factoring is a powerful technique, especially when dealing with simple quadratic equations that fit recognizable patterns like the difference of squares. However, not all quadratic equations can be easily factored. In such cases, alternative methods such as completing the square or using the quadratic formula may be more appropriate. The key to mastering factoring is to recognize common patterns and practice applying them to various quadratic equations. It is also essential to remember the zero-product property, which forms the basis for finding solutions once the equation is factored. In the next section, we will explore another method for solving the quadratic equation x² - 25 = 0, providing a different perspective and reinforcing our understanding of quadratic equation solutions. By understanding multiple methods, we equip ourselves with a versatile toolkit for tackling a wider range of problems.
Method 2: Using the Square Root Property
Another direct and efficient approach to solving the quadratic equation x² - 25 = 0 is by utilizing the square root property. This method is particularly effective when the quadratic equation can be expressed in the form x² = c, where 'c' is a constant. Our equation, x² - 25 = 0, perfectly fits this form. To apply the square root property, we first isolate the x² term. We achieve this by adding 25 to both sides of the equation: x² - 25 + 25 = 0 + 25. This simplifies to x² = 25. Now, we can apply the square root property, which states that if x² = c, then x = ±√c. This is because both the positive and negative square roots of 'c', when squared, will result in 'c'. Applying this to our equation, we take the square root of both sides: √(x²) = ±√25. The square root of x² is simply |x|, and the square root of 25 is 5. Thus, we have |x| = 5. This means that x can be either 5 or -5, as both 5² and (-5)² equal 25. Therefore, the solutions to the quadratic equation x² - 25 = 0 are x = 5 and x = -5, which aligns with the solutions we obtained through factoring. The square root property provides a concise and straightforward way to solve quadratic equations that are in the form x² = c. It avoids the need for factoring or more complex methods like completing the square or the quadratic formula. However, it's crucial to remember the ± sign when taking the square root, as both positive and negative roots satisfy the equation. This method underscores the importance of understanding the properties of square roots and their relationship to quadratic equations. It also highlights the fact that quadratic equations can have two solutions, reflecting the two points where the parabola intersects the x-axis. While the square root property is a powerful tool, it is not applicable to all quadratic equations. It is best suited for equations where the 'b' term (the coefficient of x) is zero. In cases where the 'b' term is non-zero, other methods like factoring, completing the square, or the quadratic formula may be more appropriate. In conclusion, the square root property offers a valuable shortcut for solving certain types of quadratic equations. By recognizing when this method is applicable, we can efficiently find the solutions and deepen our understanding of quadratic equations.
Choosing the Correct Answer
Having explored two different methods to solve the quadratic equation x² - 25 = 0, we have consistently arrived at the same solutions: x = 5 and x = -5. Now, let's examine the answer choices provided and identify the correct one. The answer choices are:
A. x = 5 and x = -5 B. x = 25 and x = -25 C. x = 125 and x = -125 D. no real solution
By comparing our solutions (x = 5 and x = -5) with the answer choices, it is clear that option A, x = 5 and x = -5, is the correct answer. The other options are incorrect. Option B suggests solutions of x = 25 and x = -25, but substituting these values into the original equation yields 25² - 25 = 600 ≠ 0 and (-25)² - 25 = 600 ≠ 0, so these are not valid solutions. Similarly, Option C proposes solutions of x = 125 and x = -125, which, when substituted into the equation, result in even larger discrepancies, further confirming their incorrectness. Option D,