Solving 2x^2 + 8x = X^2 - 16 A Step-by-Step Guide

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Quadratic equations are a fundamental concept in algebra, appearing in various mathematical and real-world applications. Understanding how to solve them is crucial for anyone studying mathematics, physics, engineering, or related fields. This article will provide a detailed, step-by-step guide to solving the quadratic equation 2x^2 + 8x = x^2 - 16. We'll cover the necessary algebraic manipulations, identify the quadratic form, and apply the quadratic formula to find the solutions. By the end of this article, you'll not only know the solutions to this specific equation but also have a solid foundation for tackling other quadratic equations.

Understanding Quadratic Equations

Before diving into the solution, let's first understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. This means that the highest power of the variable (usually denoted as 'x') is 2. The standard form of a quadratic equation is:

ax^2 + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. If 'a' were zero, the equation would become a linear equation instead of a quadratic one.

The coefficients 'a', 'b', and 'c' play a crucial role in determining the nature and number of solutions (also known as roots) of the quadratic equation. These roots are the values of 'x' that satisfy the equation, making the left-hand side equal to zero. Quadratic equations can have two distinct real roots, one repeated real root, or two complex roots. The nature of the roots depends on the discriminant, which we'll discuss later.

Quadratic equations appear in various contexts, from simple mathematical problems to complex scientific models. They are used to model projectile motion, calculate areas, optimize designs, and more. The ability to solve quadratic equations is therefore an essential skill in many disciplines.

Transforming the Equation into Standard Form

The given equation is 2x^2 + 8x = x^2 - 16. To solve it, we first need to transform it into the standard form ax^2 + bx + c = 0. This involves moving all terms to one side of the equation, leaving zero on the other side.

  1. Subtract x^2 from both sides:

    2x^2 + 8x - x^2 = x^2 - 16 - x^2

    This simplifies to:

    x^2 + 8x = -16

  2. Add 16 to both sides:

    x^2 + 8x + 16 = -16 + 16

    This simplifies to:

    x^2 + 8x + 16 = 0

Now, the equation is in the standard form ax^2 + bx + c = 0, where a = 1, b = 8, and c = 16. This transformation is a critical step because it allows us to directly apply methods like factoring or the quadratic formula.

Methods for Solving Quadratic Equations

There are several methods for solving quadratic equations, each with its own advantages and disadvantages. The most common methods are:

  1. Factoring: This method involves expressing the quadratic expression as a product of two linear factors. It is the simplest method when it works, but it's not always applicable.

  2. Completing the Square: This method involves manipulating the equation to form a perfect square trinomial on one side. It is a reliable method, but it can be more complex than factoring.

  3. Quadratic Formula: This method is a general formula that provides the solutions to any quadratic equation. It is the most versatile method, but it can be computationally intensive.

For the equation x^2 + 8x + 16 = 0, we can use any of these methods. However, factoring is the most straightforward approach in this case.

Solving by Factoring

Factoring involves finding two numbers that multiply to 'c' (16 in this case) and add up to 'b' (8 in this case). These numbers are 4 and 4, since 4 * 4 = 16 and 4 + 4 = 8. Therefore, we can factor the quadratic expression as follows:

x^2 + 8x + 16 = (x + 4)(x + 4)

So, the equation becomes:

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

This can also be written as:

(x + 4)^2 = 0

To find the solutions, we set each factor equal to zero:

x + 4 = 0

Solving for 'x', we get:

x = -4

Since both factors are the same, we have a repeated root. This means that the equation has only one solution, which is x = -4.

Using the Quadratic Formula

While factoring was straightforward in this case, let's also demonstrate how to use the quadratic formula to solve the same equation. The quadratic formula is given by:

x = [-b ± √(b^2 - 4ac)] / (2a)

For the equation x^2 + 8x + 16 = 0, we have a = 1, b = 8, and c = 16. Plugging these values into the formula, we get:

x = [-8 ± √(8^2 - 4 * 1 * 16)] / (2 * 1)

Simplifying the expression:

x = [-8 ± √(64 - 64)] / 2

x = [-8 ± √0] / 2

x = -8 / 2

x = -4

As we found using factoring, the quadratic formula also gives us the solution x = -4. This confirms that our solution is correct.

The term inside the square root, b^2 - 4ac, is called the discriminant. The discriminant provides valuable information about the nature of the roots:

  • If b^2 - 4ac > 0, the equation has two distinct real roots.
  • If b^2 - 4ac = 0, the equation has one repeated real root.
  • If b^2 - 4ac < 0, the equation has two complex roots.

In this case, the discriminant is 0, indicating that the equation has one repeated real root, which we found to be x = -4.

Graphical Interpretation of the Solution

Graphically, the solutions of a quadratic equation represent the x-intercepts of the parabola defined by the equation y = ax^2 + bx + c. The x-intercepts are the points where the parabola intersects the x-axis (where y = 0).

For the equation x^2 + 8x + 16 = 0, the corresponding parabola is y = x^2 + 8x + 16. Since the equation has one repeated root, the parabola touches the x-axis at only one point, which is x = -4. This point is also the vertex of the parabola.

A graph of the parabola y = x^2 + 8x + 16 would visually confirm that it touches the x-axis at x = -4, illustrating the solution we found algebraically.

Conclusion

In this article, we have provided a comprehensive guide to solving the quadratic equation 2x^2 + 8x = x^2 - 16. We transformed the equation into the standard form x^2 + 8x + 16 = 0 and solved it using both factoring and the quadratic formula. Both methods yielded the same solution, x = -4, which is a repeated real root.

We also discussed the importance of understanding quadratic equations, the different methods for solving them, and the significance of the discriminant in determining the nature of the roots. Additionally, we explored the graphical interpretation of the solution, illustrating how the roots correspond to the x-intercepts of the parabola.

Solving quadratic equations is a fundamental skill in mathematics, and mastering it opens the door to more advanced topics. By understanding the concepts and techniques discussed in this article, you can confidently tackle a wide range of quadratic equation problems.

This step-by-step guide should provide a solid foundation for solving quadratic equations. Remember to practice with different examples to enhance your understanding and problem-solving skills. Whether you prefer factoring, completing the square, or the quadratic formula, the key is to choose the method that you find most efficient and accurate for each specific equation.