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

Jul 11, 2025 by ADMIN 51 views

Solve for x in the Equation x^2 + 14x + 17 = -96

In this article, we will dive deep into solving the quadratic equation x^2 + 14x + 17 = -96. Quadratic equations are fundamental in algebra and appear in various fields of mathematics, physics, engineering, and computer science. Mastering the techniques to solve them is crucial for anyone delving into these disciplines. This comprehensive guide will walk you through the steps to find the solutions for x, providing a detailed explanation of each method used. Understanding these methods will not only help you solve this specific problem but also equip you with the tools to tackle other quadratic equations confidently. Whether you are a student, an educator, or simply someone with a keen interest in mathematics, this article aims to enhance your problem-solving skills and deepen your understanding of quadratic equations.

Before we jump into solving the equation x^2 + 14x + 17 = -96, it’s essential to understand what a quadratic equation is and its standard form. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in our 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 coefficients a, b, and c play a crucial role in determining the nature and values of the solutions (also known as roots) of the equation. Recognizing this standard form is the first step in applying appropriate methods to solve for x. In the given equation, we need to manipulate it to fit this standard form before we can apply methods such as factoring, completing the square, or using the quadratic formula. By understanding the basic structure of quadratic equations, we lay the groundwork for solving more complex problems efficiently and accurately.

The first step in solving the equation x^2 + 14x + 17 = -96 is to transform it into the standard quadratic form, ax^2 + bx + c = 0. This involves moving all terms to one side of the equation, leaving zero on the other side. To do this, we add 96 to both sides of the equation. This gives us: x^2 + 14x + 17 + 96 = 0. Simplifying the equation by combining the constant terms (17 and 96), we get: x^2 + 14x + 113 = 0. Now, the equation is in the standard quadratic form, where a = 1, b = 14, and c = 113. This transformation is crucial because it allows us to directly apply various methods for solving quadratic equations, such as the quadratic formula or completing the square. By ensuring the equation is in standard form, we set the stage for accurate and efficient solutions.

There are several methods to solve quadratic equations, each with its advantages and suitability for different types of equations. The three primary methods are:

Factoring: This method involves expressing the quadratic equation as a product of two binomials. Factoring is straightforward when the equation has integer roots, but it can be challenging for equations with complex or irrational roots.
Completing the Square: This method transforms the equation into a perfect square trinomial, making it easier to solve. Completing the square is particularly useful when the equation cannot be easily factored and provides a clear path to finding the solutions.
Quadratic Formula: This is a universal method that can solve any quadratic equation, regardless of the nature of its roots. The quadratic formula is derived from the method of completing the square and provides a direct way to find the values of x. The formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

Each method offers a unique approach to solving quadratic equations, and the choice of method often depends on the specific equation and the solver's preference. In the following sections, we will apply the quadratic formula to solve our equation, x^2 + 14x + 113 = 0.

For the equation x^2 + 14x + 113 = 0, we can use the quadratic formula to find the solutions for x. As we identified earlier, the coefficients are a = 1, b = 14, and c = 113. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / (2a)

Substituting the values of a, b, and c into the formula, we get:

x = (-14 ± √(14^2 - 4 * 1 * 113)) / (2 * 1)

Now, we simplify the expression step by step. First, calculate the square of 14:

14^2 = 196

Next, compute the product inside the square root:

4 * 1 * 113 = 452

Substitute these values back into the equation:

x = (-14 ± √(196 - 452)) / 2

Now, subtract 452 from 196:

196 - 452 = -256

So, the equation becomes:

x = (-14 ± √(-256)) / 2

The square root of a negative number involves imaginary numbers. The square root of -256 can be expressed as:

√(-256) = √(256 * -1) = √256 * √-1 = 16i

Where i is the imaginary unit, defined as √-1. Substituting this back into the equation, we get:

x = (-14 ± 16i) / 2

Now that we have x = (-14 ± 16i) / 2, we can simplify the expression by dividing both the real and imaginary parts by 2:

x = -14/2 ± (16i)/2

This simplifies to:

x = -7 ± 8i

Therefore, the solutions for x are two complex numbers:

x = -7 + 8i and x = -7 - 8i

These are the complex roots of the quadratic equation x^2 + 14x + 113 = 0. Understanding how to handle complex roots is crucial in advanced mathematics and engineering. This step-by-step simplification ensures accuracy and clarity in the solution process.

To ensure the solutions x = -7 + 8i and x = -7 - 8i are correct, we can substitute them back into the original equation, x^2 + 14x + 113 = 0, and verify that the equation holds true.

First, let's substitute x = -7 + 8i:

(-7 + 8i)^2 + 14(-7 + 8i) + 113

Expand the square:

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

Substitute this back into the equation:

(49 - 112i - 64) + 14(-7 + 8i) + 113

Distribute the 14:

(49 - 112i - 64) - 98 + 112i + 113

Combine like terms:

(49 - 64 - 98 + 113) + (-112i + 112i)

Simplify:

0 + 0i = 0

So, x = -7 + 8i is indeed a solution.

Now, let's substitute x = -7 - 8i:

(-7 - 8i)^2 + 14(-7 - 8i) + 113

Expand the square:

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

Substitute this back into the equation:

(49 + 112i - 64) + 14(-7 - 8i) + 113

Distribute the 14:

(49 + 112i - 64) - 98 - 112i + 113

Combine like terms:

(49 - 64 - 98 + 113) + (112i - 112i)

Simplify:

0 + 0i = 0

Thus, x = -7 - 8i is also a solution. This verification step is essential to confirm the accuracy of our solutions and demonstrates a thorough understanding of the problem-solving process.

In conclusion, we have successfully solved the quadratic equation x^2 + 14x + 17 = -96 by transforming it into standard form, applying the quadratic formula, and simplifying the results. The solutions we found are the complex numbers x = -7 + 8i and x = -7 - 8i. We also verified these solutions by substituting them back into the original equation, ensuring their correctness. This detailed process illustrates the importance of understanding the fundamental principles of quadratic equations and the application of appropriate methods. The quadratic formula is a powerful tool for solving any quadratic equation, and mastering its use is crucial for success in algebra and beyond. This step-by-step approach not only solves the specific problem but also provides a framework for tackling similar problems in the future. Remember, practice and a clear understanding of the underlying concepts are key to mastering quadratic equations.