Solutions Of The Equation X² - 16 = 0 A Step-by-Step Guide

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This article provides a detailed explanation of how to solve the equation x² - 16 = 0. We will explore different methods and identify the correct solutions from the given options. Understanding quadratic equations is fundamental in mathematics, and this guide will help you grasp the concepts and techniques involved.

Understanding the Problem

Before diving into the solutions, let's understand the equation x² - 16 = 0. This is a quadratic equation, which is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. In our case, a = 1, b = 0, and c = -16. Solving this equation means finding the values of x that make the equation true.

The given options for solutions are: x = -8, x = -4, x = -2, x = 2, x = 4, and x = 8. We need to check which of these values, when substituted into the equation, satisfy the equation.

Method 1: Direct Substitution

The most straightforward way to solve this is by direct substitution. We will substitute each given value of x into the equation x² - 16 = 0 and see if the result is zero. This method is reliable and helps in understanding the concept of solutions to equations. It involves a simple process of replacing the variable x with the potential solution and evaluating the expression.

  1. For x = -8:

    • Substitute x = -8 into the equation: (-8)² - 16 = 64 - 16 = 48. This does not equal zero, so x = -8 is not a solution.

  2. For x = -4:

    • Substitute x = -4 into the equation: (-4)² - 16 = 16 - 16 = 0. This equals zero, so x = -4 is a solution.

  3. For x = -2:

    • Substitute x = -2 into the equation: (-2)² - 16 = 4 - 16 = -12. This does not equal zero, so x = -2 is not a solution.

  4. For x = 2:

    • Substitute x = 2 into the equation: (2)² - 16 = 4 - 16 = -12. This does not equal zero, so x = 2 is not a solution.

  5. For x = 4:

    • Substitute x = 4 into the equation: (4)² - 16 = 16 - 16 = 0. This equals zero, so x = 4 is a solution.

  6. For x = 8:

    • Substitute x = 8 into the equation: (8)² - 16 = 64 - 16 = 48. This does not equal zero, so x = 8 is not a solution.

From this method, we find that x = -4 and x = 4 are the solutions to the equation.

Method 2: Factoring

Another efficient method to solve the equation x² - 16 = 0 is by factoring. This method involves rewriting the equation as a product of simpler expressions. Recognizing algebraic patterns is crucial for this method, and it provides a deeper understanding of the equation's structure. Factoring is a powerful technique that simplifies the process of finding solutions.

The equation x² - 16 is a difference of squares, which can be factored using the formula a² - b² = (a - b)(a + b). In our case, a = x and b = 4, so we can rewrite the equation as:

x² - 16 = (x - 4)(x + 4)

Now, we set the factored expression equal to zero:

(x - 4)(x + 4) = 0

For this product to be zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x:

  1. x - 4 = 0

    • Adding 4 to both sides gives x = 4.

  2. x + 4 = 0

    • Subtracting 4 from both sides gives x = -4.

Thus, the solutions are x = 4 and x = -4, which aligns with the solutions we found using the direct substitution method.

Method 3: Using the Square Root Property

The square root property is another effective method for solving equations of the form x² = k, where k is a constant. This method is based on the principle that if the square of a number is equal to a constant, then the number itself is equal to the square root of the constant or its negative. Applying this property requires a clear understanding of the relationship between squares and square roots.

We start with the equation:

x² - 16 = 0

Add 16 to both sides to isolate the x² term:

x² = 16

Now, take the square root of both sides:

√x² = ±√16

This gives us:

x = ±4

So, the solutions are x = 4 and x = -4. This method provides a quick way to solve such equations and reinforces the concept of both positive and negative roots.

Graphical Interpretation

Visualizing the equation x² - 16 = 0 graphically can provide additional insights. The equation represents a parabola when plotted on a coordinate plane. The solutions to the equation are the points where the parabola intersects the x-axis. This graphical approach helps in understanding the nature of the solutions and their relationship to the equation.

Consider the function f(x) = x² - 16. The graph of this function is a parabola that opens upwards. The points where the parabola intersects the x-axis are the solutions to the equation f(x) = 0. These points are also known as the roots or zeros of the function.

By plotting the graph, we can see that the parabola intersects the x-axis at x = -4 and x = 4. This graphical representation confirms the solutions we found using the algebraic methods.

Importance of Checking Solutions

While we have identified potential solutions using different methods, it is always crucial to check the solutions by substituting them back into the original equation. This step ensures that the solutions are correct and that no errors were made during the solving process. Checking solutions is a fundamental practice in mathematics and helps in verifying the accuracy of the results.

We have already performed this check in Method 1 (Direct Substitution), but it's worth emphasizing the importance of this step. Substituting the solutions back into the original equation confirms that they indeed satisfy the equation.

Common Mistakes to Avoid

When solving quadratic equations, it is essential to avoid common mistakes that can lead to incorrect solutions. One frequent error is forgetting to consider both the positive and negative square roots when using the square root property. Another common mistake is incorrect factoring or not factoring completely. Being aware of these pitfalls helps in solving equations accurately.

  1. Forgetting the negative root: When taking the square root of both sides of an equation, remember to include both the positive and negative roots. For example, when solving x² = 16, both x = 4 and x = -4 are valid solutions.

  2. Incorrect factoring: Ensure that the factoring is done correctly. Double-check the factored form by expanding it to see if it matches the original expression.

  3. Dividing by a variable: Avoid dividing both sides of an equation by a variable term, as this may lead to losing a solution. For example, if you have x² = 4x, dividing by x would give x = 4, but you would miss the solution x = 0.

Real-World Applications

Quadratic equations have numerous real-world applications in various fields, including physics, engineering, economics, and computer science. They are used to model projectile motion, calculate areas and volumes, optimize processes, and solve many other practical problems. Understanding how to solve quadratic equations is therefore essential for many careers and everyday situations.

For instance, in physics, quadratic equations are used to describe the trajectory of a projectile, such as a ball thrown into the air. In engineering, they are used in the design of structures and circuits. In economics, they can be used to model supply and demand curves. These applications highlight the importance of mastering the techniques for solving quadratic equations.

Conclusion

In conclusion, the solutions to the equation x² - 16 = 0 are x = -4 and x = 4. We have demonstrated this using three different methods: direct substitution, factoring, and the square root property. Understanding these methods and their applications is crucial for solving quadratic equations effectively. By avoiding common mistakes and checking solutions, you can ensure accuracy in your calculations. Quadratic equations are a fundamental concept in mathematics with wide-ranging applications in various fields, making it essential to have a solid grasp of their solutions.