Solving X In The Equation X^2 + 14x + 17 = -96
Solving quadratic equations is a fundamental skill in algebra, and this article will guide you through the process of finding the solutions for the equation x^2 + 14x + 17 = -96. This particular equation might seem daunting at first glance, but by using the right techniques, such as completing the square or applying the quadratic formula, we can systematically find the values of x that satisfy the equation. Understanding how to solve such equations is crucial for various applications in mathematics, physics, engineering, and other fields.
Understanding Quadratic Equations
Before diving into the solution, let's briefly discuss what quadratic equations are and the methods we can use to solve them. 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 are also known as roots or zeros.
There are several methods to solve quadratic equations:
-
Factoring: If the quadratic expression can be factored easily, this method is often the quickest. We look for two numbers that multiply to c and add to b. However, not all quadratic equations can be factored easily.
-
Completing the Square: This method involves transforming the equation into a perfect square trinomial. It is a powerful method that can be used to solve any quadratic equation, but it can be a bit more involved.
-
Quadratic Formula: This formula provides a direct way to find the solutions. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
This formula works for all quadratic equations, regardless of whether they can be factored or not.
-
Graphical Method: While not as precise as algebraic methods, graphing the quadratic equation can give a visual representation of the solutions. The roots are the points where the parabola intersects the x-axis.
For this particular equation, the quadratic formula and completing the square are the most suitable methods due to the presence of a constant term that makes factoring less straightforward.
Method 1: Using the Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form ax^2 + bx + c = 0. In our case, the equation is x^2 + 14x + 17 = -96. The first step is to rewrite the equation in the standard form by adding 96 to both sides:
x^2 + 14x + 17 + 96 = 0
This simplifies to:
x^2 + 14x + 113 = 0
Now we can identify the coefficients:
- a = 1
- b = 14
- c = 113
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, we get:
x = (-14 ± √(14^2 - 4 * 1 * 113)) / (2 * 1)
Now, let's simplify the expression step by step. First, calculate the value inside the square root:
14^2 - 4 * 1 * 113 = 196 - 452 = -256
So the equation becomes:
x = (-14 ± √(-256)) / 2
Since we have a negative number inside the square root, the solutions will be complex numbers. Recall that √(-1) = i, where i is the imaginary unit. We can rewrite √(-256) as:
√(-256) = √(256 * -1) = √(256) * √(-1) = 16i
Now substitute this back into the equation:
x = (-14 ± 16i) / 2
Divide both terms in the numerator by 2:
x = -7 ± 8i
Thus, the solutions are:
x = -7 + 8i and x = -7 - 8i
Therefore, the solutions to the equation x^2 + 14x + 17 = -96 are complex numbers, specifically -7 + 8i and -7 - 8i. This method demonstrates the power of the quadratic formula in handling equations with complex roots. Understanding each step, from rewriting the equation to simplifying the square root of a negative number, is crucial for mastering quadratic equations.
Method 2: Completing the Square
Completing the square is another powerful technique to solve quadratic equations. This method is particularly useful when the equation doesn't factor easily, and it provides a way to rewrite the quadratic in a form that allows us to isolate x. Let's apply this method to the equation x^2 + 14x + 17 = -96.
The first step, as before, is to rewrite the equation in the standard form by adding 96 to both sides:
x^2 + 14x + 17 + 96 = 0
This simplifies to:
x^2 + 14x + 113 = 0
Now, we want to rewrite the left side of the equation as a perfect square trinomial. A perfect square trinomial is of the form (x + k)^2 = x^2 + 2kx + k^2. To complete the square, we need to add and subtract a value that makes the expression a perfect square. In our equation, the coefficient of x is 14. We take half of this coefficient and square it:
(14 / 2)^2 = 7^2 = 49
So, we add and subtract 49 to the left side of the equation:
x^2 + 14x + 49 - 49 + 113 = 0
Now, we can group the first three terms, which form a perfect square:
(x^2 + 14x + 49) + (-49 + 113) = 0
The expression in the parentheses is (x + 7)^2. Simplify the remaining terms:
(x + 7)^2 + 64 = 0
Now, we isolate the squared term by subtracting 64 from both sides:
(x + 7)^2 = -64
To solve for x, we take the square root of both sides:
√( (x + 7)^2 ) = ±√(-64)
This gives us:
x + 7 = ±√(-64)
Since we have a negative number inside the square root, we again encounter complex numbers. We can rewrite √(-64) as:
√(-64) = √(64 * -1) = √(64) * √(-1) = 8i
So the equation becomes:
x + 7 = ±8i
Now, subtract 7 from both sides to solve for x:
x = -7 ± 8i
Thus, the solutions are:
x = -7 + 8i and x = -7 - 8i
These are the same solutions we obtained using the quadratic formula. Completing the square is a valuable method not only for solving quadratic equations but also for understanding the structure of quadratic expressions and their graphical representations. The process involves transforming the equation into a form where the variable can be easily isolated, providing a clear path to the solutions.
Verifying the Solutions
To ensure the accuracy of our solutions, it's always a good practice to verify them by plugging them back into the original equation. We found the solutions to be x = -7 + 8i and x = -7 - 8i. The original equation is x^2 + 14x + 17 = -96.
Let's verify the first solution, x = -7 + 8i:
(-7 + 8i)^2 + 14(-7 + 8i) + 17
First, expand (-7 + 8i)^2:
(-7 + 8i)^2 = (-7 + 8i)(-7 + 8i) = 49 - 56i - 56i + 64i^2
Since i^2 = -1, we have:
49 - 112i - 64 = -15 - 112i
Now, substitute this back into the equation:
-15 - 112i + 14(-7 + 8i) + 17
Distribute the 14:
-15 - 112i - 98 + 112i + 17
Combine the real and imaginary parts:
(-15 - 98 + 17) + (-112i + 112i) = -96 + 0i = -96
Since the result is -96, the first solution x = -7 + 8i is correct.
Now, let's verify the second solution, x = -7 - 8i:
(-7 - 8i)^2 + 14(-7 - 8i) + 17
First, expand (-7 - 8i)^2:
(-7 - 8i)^2 = (-7 - 8i)(-7 - 8i) = 49 + 56i + 56i + 64i^2
Since i^2 = -1, we have:
49 + 112i - 64 = -15 + 112i
Now, substitute this back into the equation:
-15 + 112i + 14(-7 - 8i) + 17
Distribute the 14:
-15 + 112i - 98 - 112i + 17
Combine the real and imaginary parts:
(-15 - 98 + 17) + (112i - 112i) = -96 + 0i = -96
Since the result is -96, the second solution x = -7 - 8i is also correct.
Therefore, both solutions x = -7 + 8i and x = -7 - 8i are verified to be correct. This process of verification is a critical step in problem-solving, especially when dealing with complex numbers and quadratic equations. It ensures that our algebraic manipulations have been accurate and that the solutions we have found are indeed valid.
Conclusion
In this article, we tackled the problem of solving the quadratic equation x^2 + 14x + 17 = -96. We employed two primary methods: the quadratic formula and completing the square. Both methods led us to the same solutions: x = -7 + 8i and x = -7 - 8i. These solutions are complex numbers, which arise due to the negative discriminant in the quadratic formula.
The quadratic formula provides a direct and reliable way to find the solutions of any quadratic equation, regardless of its factorability. By identifying the coefficients a, b, and c, and substituting them into the formula, we can systematically find the roots.
Completing the square is another powerful technique that not only helps in solving quadratic equations but also enhances our understanding of the structure of quadratic expressions. This method involves rewriting the quadratic equation in a form where a perfect square trinomial is present, allowing us to isolate the variable and find the solutions.
We also emphasized the importance of verifying the solutions by substituting them back into the original equation. This step is crucial to ensure that our algebraic manipulations are correct and that the solutions are valid.
Solving quadratic equations is a fundamental skill in algebra with wide-ranging applications in various fields. Mastering different methods to solve these equations equips us with the tools to tackle more complex mathematical problems and real-world applications. Whether it's physics, engineering, or computer science, the ability to solve quadratic equations is a valuable asset. Understanding the underlying principles and practicing these methods will solidify your grasp of algebra and enhance your problem-solving abilities.
The solutions to the equation x^2 + 14x + 17 = -96 are indeed x = -7 ± 8i, showcasing the power and versatility of algebraic techniques in solving mathematical problems.
