Solving For X In The Equation X^2 + 14x + 17 = -96

Finding the solutions to quadratic equations is a fundamental concept in algebra. In this article, we will delve into the process of solving for $x$ in the equation $x^2 + 14x + 17 = -96$. This equation, a classic example of a quadratic equation, requires a methodical approach to arrive at the correct solutions. We will explore the steps involved, from simplifying the equation to applying the quadratic formula, and discuss the nature of the roots obtained. This detailed explanation aims to provide clarity and a deeper understanding of solving quadratic equations.

Understanding Quadratic Equations

Before diving into the solution, it's crucial to understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. 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 zero. The solutions to a quadratic equation, also known as roots or zeros, are the values of $x$ that satisfy the equation. These roots can be real or complex numbers, depending on the discriminant ($b^2 - 4ac$) of the equation.

In the given equation, $x^2 + 14x + 17 = -96$, we can identify the coefficients as follows: the coefficient of $x^2$ is 1, the coefficient of $x$ is 14, and the constant term is initially 17. However, to solve the equation, we must first rewrite it in the standard form $ax^2 + bx + c = 0$. This involves moving the constant term on the right side of the equation to the left side, which we will do in the next step.

Understanding the nature of quadratic equations and their components is essential for choosing the appropriate method to solve them. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. The choice of method often depends on the specific form of the equation and the ease with which it can be manipulated. In this case, we will primarily use the quadratic formula, as it is a versatile method that can be applied to any quadratic equation, regardless of the nature of its roots.

Step 1: Simplify the Equation

The first step in solving the equation $x^2 + 14x + 17 = -96$ is to rewrite it in the standard quadratic form, which is $ax^2 + bx + c = 0$. To do this, we need to move the constant term from the right side of the equation to the left side. This involves adding 96 to both sides of the equation.

Adding 96 to both sides, we get:

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

Simplifying this gives us:

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

Now, we have the equation in the standard quadratic form, where $a = 1$, $b = 14$, and $c = 113$. This form is crucial because it allows us to easily identify the coefficients needed for the quadratic formula, which is our next step in finding the solutions for $x$.

Rewriting the equation in the standard form not only makes it easier to apply the quadratic formula but also helps in visualizing the equation in a more structured manner. This step is essential for accurately identifying the coefficients and ensuring that the quadratic formula is applied correctly. A clear understanding of this initial step is fundamental to solving any quadratic equation effectively.

Step 2: Apply the Quadratic Formula

With the equation now in the standard form $x^2 + 14x + 113 = 0$, we can apply the quadratic formula to solve for $x$. The quadratic formula is a powerful tool that provides the solutions to any quadratic equation of the form $ax^2 + bx + c = 0$. The formula is given by:

$x = \frac{-b pm sqrt{b^2 - 4ac}}{2a}$

In our equation, we have $a = 1$, $b = 14$, and $c = 113$. Substituting these values into the quadratic formula, we get:

$x = \frac{-14 pm sqrt{14^2 - 4(1)(113)}}{2(1)}$

Now, we need to simplify the expression under the square root and the rest of the formula. This involves performing the arithmetic operations and carefully handling the signs. The next step is to calculate the discriminant, which is the term inside the square root ($b^2 - 4ac$). The discriminant will tell us about the nature of the roots – whether they are real or complex, and whether they are distinct or repeated.

The quadratic formula is a cornerstone of algebra, and its application is not limited to simple quadratic equations. It is a universal method that can be used to solve any quadratic equation, regardless of the complexity of its coefficients. Understanding and correctly applying this formula is a critical skill in algebra and beyond.

Step 3: Simplify the Expression

Now that we have applied the quadratic formula, we need to simplify the expression to find the values of $x$. Recall that we have:

$x = \frac{-14 pm sqrt{14^2 - 4(1)(113)}}{2(1)}$

First, let's simplify the expression under the square root, which is the discriminant:

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

Since the discriminant is negative, this indicates that the solutions for $x$ will be complex numbers. Now, we can substitute this value back into the quadratic formula:

$x = \frac{-14 pm sqrt{-256}}{2}$

To simplify the square root of a negative number, we use the imaginary unit $i$, where $i = sqrt{-1}$. Therefore, $ sqrt{-256} = sqrt{256} sqrt{-1} = 16i$. Substituting this back into the equation, we get:

$x = \frac{-14 pm 16i}{2}$

Now, we can divide both the real and imaginary parts of the numerator by the denominator:

$x = -7 pm 8i$

This gives us two complex solutions for $x$. The simplification process involves careful arithmetic and an understanding of complex numbers, particularly the imaginary unit $i$. The ability to simplify such expressions is crucial for arriving at the final solutions of the quadratic equation.

Step 4: Identify the Solutions

After simplifying the expression obtained from the quadratic formula, we have arrived at the solutions for $x$:

$x = -7 pm 8i$

This expression represents two complex solutions: one with the plus sign and one with the minus sign. These solutions are:

1. $x = -7 + 8i$
2. $x = -7 - 8i$

These are complex conjugate pairs, which is a common occurrence when solving quadratic equations with a negative discriminant. Complex conjugate pairs have the form $a + bi$ and $a - bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

Therefore, the solutions to the equation $x^2 + 14x + 17 = -96$ are $x = -7 + 8i$ and $x = -7 - 8i$. Comparing these solutions with the given options, we can see that they match option B.

Identifying the solutions involves understanding the notation and the nature of complex numbers. The ability to correctly interpret the solutions is essential for answering the question accurately and for further applications in mathematics and related fields. In this case, the solutions are complex, indicating that the parabola represented by the quadratic equation does not intersect the x-axis.

Conclusion

In conclusion, solving the quadratic equation $x^2 + 14x + 17 = -96$ involves several key steps. First, we rewrite the equation in the standard form $ax^2 + bx + c = 0$, which gives us $x^2 + 14x + 113 = 0$. Then, we apply the quadratic formula, $x = \frac{-b pm sqrt{b^2 - 4ac}}{2a}$, to find the solutions for $x$. Substituting the values $a = 1$, $b = 14$, and $c = 113$ into the formula, we simplify the expression and find that the discriminant is negative, indicating complex solutions.

After simplifying the expression, we arrive at the solutions $x = -7 pm 8i$, which are complex conjugate pairs. This means the correct answer is option B: $x = -7 pm 8i$. This process demonstrates the importance of understanding quadratic equations, the quadratic formula, and complex numbers. It also highlights the need for careful arithmetic and algebraic manipulation to arrive at the correct solutions.

Solving quadratic equations is a fundamental skill in algebra, and mastering this process is essential for success in higher-level mathematics. The steps outlined in this article provide a comprehensive guide to solving such equations, and the example demonstrates the application of these steps in a specific case. By following these steps, students and practitioners can confidently solve quadratic equations and understand the nature of their solutions.