Solving $x^2 + 14x + 17 = -96$ A Comprehensive Guide

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Introduction

In this comprehensive guide, we will delve into the intricacies of solving quadratic equations, specifically focusing on the equation $x^2 + 14x + 17 = -96$. Quadratic equations are a fundamental concept in algebra, and mastering their solutions is crucial for various mathematical and scientific applications. This article provides a step-by-step approach to solving this particular equation, exploring different methods and highlighting key concepts. Whether you're a student looking to enhance your algebra skills or a math enthusiast seeking a refresher, this guide will equip you with the knowledge and techniques needed to tackle similar problems. We will address the nuances of complex solutions, ensuring a thorough understanding of the subject matter.

Understanding Quadratic Equations

Before diving into the solution, it's essential to understand the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this case, 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, which represent the values of x that satisfy the equation.

Quadratic equations can have two distinct real roots, one real root (a repeated root), or two complex roots. The nature of the roots depends on the discriminant, which is given by the formula $D = b^2 - 4ac$. If D > 0, there are two distinct real roots; if D = 0, there is one real root; and if D < 0, there are two complex roots. Complex roots involve the imaginary unit i, where $i^2 = -1$. Understanding these concepts is crucial for solving quadratic equations effectively and interpreting their solutions.

Transforming the Equation

The first step in solving the given equation, $x^2 + 14x + 17 = -96$, is to rewrite it in the standard form of a quadratic equation, $ax^2 + bx + c = 0$. To do this, we need to add 96 to both sides of the equation:

$x^2 + 14x + 17 + 96 = -96 + 96$

This simplifies to:

$x^2 + 14x + 113 = 0$

Now, we have the equation in the standard form, where a = 1, b = 14, and c = 113. This form allows us to apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Each method has its advantages and is suitable for different types of quadratic equations. The next sections will explore these methods and determine the most efficient approach for solving our specific equation.

The Quadratic Formula

The most versatile method for solving quadratic equations is the quadratic formula. This formula provides a direct way to find the roots of any quadratic equation, regardless of whether the roots are real or complex. The quadratic formula is derived from the process of completing the square and is given by:

x = rac{-b ext{ ± } ext{√}(b^2 - 4ac)}{2a}

Where a, b, and c are the coefficients of the quadratic equation $ax^2 + bx + c = 0$. In our case, we have a = 1, b = 14, and c = 113. Plugging these values into the quadratic formula, we get:

x = rac{-14 ext{ ± } ext{√}(14^2 - 4 imes 1 imes 113)}{2 imes 1}

Now, we need to simplify the expression under the square root:

$14^2 - 4 imes 1 imes 113 = 196 - 452 = -256$

So, our equation becomes:

x = rac{-14 ext{ ± } ext{√}(-256)}{2}

Since the value under the square root is negative, we will have complex roots. We know that the square root of -1 is the imaginary unit i, so we can rewrite the square root of -256 as:

$ ext{√}(-256) = ext{√}(256 imes -1) = ext{√}256 imes ext{√}-1 = 16i$

Calculating the Complex Solutions

Now that we have simplified the square root, we can substitute it back into the quadratic formula:

x = rac{-14 ext{ ± } 16i}{2}

To find the two complex solutions, we need to consider both the plus and minus signs. We can divide both terms in the numerator by 2:

$x = -7 ext{ ± } 8i$

This gives us two complex solutions:

$x_1 = -7 + 8i$ $x_2 = -7 - 8i$

These are the complex roots of the quadratic equation $x^2 + 14x + 113 = 0$. Complex roots always come in conjugate pairs, meaning they have the same real part but opposite imaginary parts. This is evident in our solutions, where -7 is the real part, and 8i and -8i are the imaginary parts.

Verifying the Solutions

To ensure the accuracy of our solutions, it's crucial to verify them by substituting them back into the original equation. Let's substitute $x = -7 + 8i$ into the equation $x^2 + 14x + 17 = -96$:

$(-7 + 8i)^2 + 14(-7 + 8i) + 17 = ? -96$

First, we expand the squared term:

$(-7 + 8i)^2 = (-7 + 8i)(-7 + 8i) = 49 - 56i - 56i + 64i^2 = 49 - 112i - 64$ (since $i^2 = -1$)

This simplifies to:

$-15 - 112i$

Next, we multiply 14 by (-7 + 8i):

$14(-7 + 8i) = -98 + 112i$

Now, we substitute these results back into the equation:

$(-15 - 112i) + (-98 + 112i) + 17 = -15 - 98 + 17 = -96$

The left side of the equation equals -96, which matches the right side. This confirms that $x = -7 + 8i$ is a valid solution.

We can perform a similar verification for $x = -7 - 8i$:

$(-7 - 8i)^2 + 14(-7 - 8i) + 17 = ? -96$

$(-7 - 8i)^2 = (-7 - 8i)(-7 - 8i) = 49 + 56i + 56i + 64i^2 = 49 + 112i - 64 = -15 + 112i$

$14(-7 - 8i) = -98 - 112i$

Substituting back into the equation:

$(-15 + 112i) + (-98 - 112i) + 17 = -15 - 98 + 17 = -96$

This also equals -96, confirming that $x = -7 - 8i$ is a valid solution. Therefore, both complex solutions satisfy the original equation.

Alternative Methods: Completing the Square

While the quadratic formula is a reliable method for solving quadratic equations, it's also beneficial to explore alternative techniques such as completing the square. Completing the square involves manipulating the quadratic equation into a perfect square trinomial, which can then be easily solved. This method not only provides the solutions but also enhances understanding of the structure of quadratic equations.

To solve $x^2 + 14x + 113 = 0$ by completing the square, we first isolate the terms involving x:

$x^2 + 14x = -113$

Next, we need to add a value to both sides of the equation to make the left side a perfect square trinomial. The value we add is the square of half the coefficient of the x term. In this case, the coefficient of x is 14, so half of it is 7, and the square of 7 is 49. Adding 49 to both sides, we get:

$x^2 + 14x + 49 = -113 + 49$

The left side is now a perfect square trinomial, which can be factored as:

$(x + 7)^2 = -64$

Now, we take the square root of both sides:

$x + 7 = ext{±} ext{√}(-64)$

Since the square root of -64 is 8i, we have:

$x + 7 = ext{±} 8i$

Finally, we subtract 7 from both sides to solve for x:

$x = -7 ext{ ± } 8i$

This method yields the same complex solutions as the quadratic formula, confirming the accuracy of our results. Completing the square is a valuable technique, especially for understanding the derivation of the quadratic formula and for solving specific types of quadratic equations where factoring is not straightforward.

Conclusion

In this detailed guide, we have successfully solved the quadratic equation $x^2 + 14x + 17 = -96$, demonstrating the application of the quadratic formula and the method of completing the square. We found that the solutions are complex numbers, specifically $x = -7 + 8i$ and $x = -7 - 8i$. The process involved rewriting the equation in standard form, applying the quadratic formula, simplifying complex numbers, and verifying the solutions. Additionally, we explored completing the square as an alternative method, reinforcing the understanding of quadratic equation solutions.

Mastering the techniques for solving quadratic equations is essential for success in algebra and beyond. Quadratic equations appear in various contexts, from physics and engineering to economics and computer science. A strong grasp of these concepts enables you to tackle more complex problems and build a solid foundation in mathematics. By understanding the nature of roots, the quadratic formula, and methods like completing the square, you can confidently approach any quadratic equation and find its solutions, whether they are real or complex.

This article has provided a comprehensive approach to solving quadratic equations with complex solutions. By following the steps and understanding the underlying principles, you can enhance your problem-solving skills and deepen your appreciation for the elegance and power of mathematics. Keep practicing and exploring different types of quadratic equations to further solidify your understanding and mastery of the subject.