Solving X² - 16 = 0 Find The Correct Solutions
In the realm of mathematics, quadratic equations hold a fundamental position, appearing in various fields such as physics, engineering, and economics. Understanding how to solve these equations is crucial for anyone seeking to delve deeper into these disciplines. This article will provide a comprehensive guide to solving the quadratic equation x² - 16 = 0, exploring different methods and highlighting the key concepts involved. We will meticulously examine each step to ensure clarity and comprehension, ultimately leading you to the correct solutions.
The quadratic equation we are tackling today is x² - 16 = 0. This equation is a special case of a quadratic equation, where the linear term (the term with 'x') is absent. This form simplifies the solution process, allowing us to use straightforward algebraic manipulations to find the roots. The roots of an equation are the values of the variable (in this case, 'x') that make the equation true. In simpler terms, they are the points where the graph of the equation intersects the x-axis. Solving a quadratic equation means finding these roots.
Method 1: Factoring
One of the most common and efficient methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic expression as a product of two linear expressions. This method is particularly effective when the quadratic equation can be easily factored. In our case, x² - 16 = 0 can be recognized as a difference of squares. The difference of squares pattern is a fundamental algebraic identity that states a² - b² = (a + b)(a - b). Recognizing this pattern is key to efficiently factoring the equation.
Applying the difference of squares pattern to our equation, we can rewrite x² - 16 as x² - 4². Now, we can directly apply the identity: x² - 4² = (x + 4)(x - 4). Therefore, the equation x² - 16 = 0 can be rewritten as (x + 4)(x - 4) = 0. This factored form of the equation is crucial because it allows us to use the zero-product property. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. This property is the cornerstone of solving equations by factoring.
Using the zero-product property, we set each factor equal to zero: x + 4 = 0 and x - 4 = 0. Solving the first equation, x + 4 = 0, we subtract 4 from both sides to isolate x, resulting in x = -4. Solving the second equation, x - 4 = 0, we add 4 to both sides to isolate x, resulting in x = 4. Thus, the solutions to the equation x² - 16 = 0 are x = -4 and x = 4. Factoring provides a clear and concise way to arrive at the solutions, especially when the equation fits a recognizable pattern like the difference of squares.
Method 2: Square Root Property
Another effective method for solving quadratic equations, especially those in the form x² = c (where c is a constant), is the square root property. This property states that if x² = c, then x = ±√c. The square root property is a direct consequence of the definition of a square root and provides a shortcut for solving equations where the variable is squared and isolated.
To apply the square root property to our equation, x² - 16 = 0, we first need to isolate the x² term. We achieve this by adding 16 to both sides of the equation, resulting in x² = 16. Now that we have the equation in the form x² = c, we can apply the square root property. Taking the square root of both sides of the equation x² = 16, we get x = ±√16. It's crucial to remember the ± sign because both positive and negative square roots will satisfy the original equation. The square root of 16 is 4, so we have x = ±4.
This means that x can be either +4 or -4. Therefore, the solutions to the equation x² - 16 = 0 are x = 4 and x = -4. The square root property offers a straightforward and efficient way to solve quadratic equations that are in the form x² = c. It avoids the need for factoring and directly leads to the solutions, making it a valuable tool in solving quadratic equations.
Method 3: Quadratic Formula (General Approach)
While the methods of factoring and the square root property are efficient for specific types of quadratic equations, the quadratic formula provides a general solution for any quadratic equation in the form ax² + bx + c = 0. The quadratic formula is a powerful tool that guarantees a solution, regardless of whether the equation can be easily factored or solved using the square root property. The formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
To apply the quadratic formula, we first need to identify the coefficients a, b, and c in our equation, x² - 16 = 0. Comparing this equation to the standard form ax² + bx + c = 0, we can see that a = 1, b = 0 (since there is no x term), and c = -16. Now that we have identified the coefficients, we can substitute them into the quadratic formula.
Substituting a = 1, b = 0, and c = -16 into the quadratic formula, we get:
x = (-0 ± √(0² - 4 * 1 * -16)) / (2 * 1)
Simplifying the expression under the square root, we have:
x = (± √(64)) / 2
The square root of 64 is 8, so we get:
x = (± 8) / 2
Now we can find the two possible values for x:
x = 8 / 2 = 4
x = -8 / 2 = -4
Thus, the solutions to the equation x² - 16 = 0 are x = 4 and x = -4. The quadratic formula, while more complex than factoring or the square root property, is a versatile method that can solve any quadratic equation. It is particularly useful when the equation does not factor easily or when the square root property is not directly applicable. Understanding and applying the quadratic formula is a fundamental skill in algebra.
Verification of Solutions
After finding the solutions to an equation, it is always a good practice to verify that the solutions are correct. This step helps to ensure that no errors were made during the solving process and that the solutions indeed satisfy the original equation. To verify the solutions, we substitute each solution back into the original equation and check if the equation holds true.
Let's start by verifying the solution x = 4. Substituting x = 4 into the equation x² - 16 = 0, we get:
4² - 16 = 16 - 16 = 0
Since the equation holds true, x = 4 is indeed a solution. Now, let's verify the solution x = -4. Substituting x = -4 into the equation x² - 16 = 0, we get:
(-4)² - 16 = 16 - 16 = 0
Again, the equation holds true, confirming that x = -4 is also a solution. Therefore, we have verified that both x = 4 and x = -4 are solutions to the equation x² - 16 = 0. Verification is a crucial step in problem-solving, as it helps to catch any potential errors and build confidence in the accuracy of the solutions.
Conclusion
In conclusion, we have explored three different methods for solving the quadratic equation x² - 16 = 0: factoring, the square root property, and the quadratic formula. Each method offers a unique approach to solving quadratic equations, and the choice of method often depends on the specific form of the equation. Factoring is efficient when the equation can be easily factored, the square root property is ideal for equations in the form x² = c, and the quadratic formula provides a general solution for any quadratic equation. By understanding and mastering these methods, you can confidently solve a wide range of quadratic equations.
The solutions to the equation x² - 16 = 0 are x = 4 and x = -4. These solutions represent the points where the graph of the equation intersects the x-axis. Verifying the solutions by substituting them back into the original equation confirms their accuracy. Solving quadratic equations is a fundamental skill in mathematics, and the methods discussed in this article provide a solid foundation for further exploration of mathematical concepts. Whether you are a student learning algebra or someone looking to refresh your mathematical skills, understanding how to solve quadratic equations is essential for success in various fields.
