Solving For X In The Equation X^2 + 11x + 121/4 = 125/4 A Step-by-Step Guide

In this comprehensive guide, we will delve into the process of solving for x in the quadratic equation $x^2 + 11x + \frac{121}{4} = \frac{125}{4}$. This equation, while seemingly complex, can be tackled systematically using fundamental algebraic principles. We will explore various methods, including completing the square, to arrive at the solution. Understanding how to solve such equations is crucial in various fields, including mathematics, physics, engineering, and computer science. Quadratic equations often arise in modeling real-world phenomena, such as projectile motion, optimization problems, and circuit analysis. Therefore, mastering the techniques for solving them is an invaluable skill.

1. Understanding Quadratic Equations

Before diving into the solution, let's establish a clear understanding 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 ≠ 0. The equation we are addressing, $x^2 + 11x + \frac{121}{4} = \frac{125}{4}$, fits this form, although it requires a slight rearrangement to match the standard form perfectly. Recognizing the structure of a quadratic equation is the first step toward solving it. It allows us to choose the most appropriate method for finding the roots, which are the values of x that satisfy the equation. These roots represent the points where the parabola described by the quadratic equation intersects the x-axis on a graph.

2. Simplifying the Equation

Our initial equation is $x^2 + 11x + \frac{121}{4} = \frac{125}{4}$. To simplify this, the first step is to eliminate the fraction on the right-hand side by subtracting $\frac{125}{4}$ from both sides. This gives us: $x^2 + 11x + \frac{121}{4} - \frac{125}{4} = 0$. Now, we can combine the constant terms on the left side: $x^2 + 11x - \frac{4}{4} = 0$. This simplifies further to $x^2 + 11x - 1 = 0$. This simplified form is much easier to work with and clearly presents the quadratic equation in the standard form $ax^2 + bx + c = 0$, where a = 1, b = 11, and c = -1. By simplifying the equation, we have prepared it for the application of various solution methods, such as completing the square or using the quadratic formula.

3. Completing the Square Method

Completing the square is a powerful technique for solving quadratic equations. It involves transforming the equation into a perfect square trinomial, which can then be easily factored. In our simplified equation, $x^2 + 11x - 1 = 0$, we will focus on the $x^2 + 11x$ terms. To complete the square, we need to add and subtract $(\frac{b}{2})^2$ to the left side of the equation, where b is the coefficient of the x term. In this case, b = 11, so $(\frac{b}{2})^2 = (\frac{11}{2})^2 = \frac{121}{4}$. Adding and subtracting this value, we get: $x^2 + 11x + \frac{121}{4} - \frac{121}{4} - 1 = 0$. Now, we can rewrite the first three terms as a perfect square: $(x + \frac{11}{2})^2 - \frac{121}{4} - 1 = 0$. Simplifying further, we have $(x + \frac{11}{2})^2 - \frac{125}{4} = 0$. This form allows us to isolate the squared term and proceed to solve for x.

4. Isolating the Squared Term

Having completed the square, our equation now stands as $(x + \frac{11}{2})^2 - \frac{125}{4} = 0$. The next crucial step is to isolate the squared term. We achieve this by adding $\frac{125}{4}$ to both sides of the equation, which results in $(x + \frac{11}{2})^2 = \frac{125}{4}$. This step is vital because it sets the stage for taking the square root of both sides, which is the key to unraveling the value of x. Isolating the squared term allows us to treat the expression $(x + \frac{11}{2})$ as a single entity, making the subsequent steps in the solution process more straightforward. This isolation technique is a fundamental aspect of solving equations involving squared terms.

5. Taking the Square Root

With the squared term isolated, we have $(x + \frac{11}{2})^2 = \frac{125}{4}$. Now, we take the square root of both sides of the equation. It's essential to remember that taking the square root yields both positive and negative solutions. So, we get $x + \frac{11}{2} = ±\sqrt{\frac{125}{4}}$. We can simplify the square root of $\frac{125}{4}$ as follows: $\sqrt{\frac{125}{4}} = \frac{\sqrt{125}}{\sqrt{4}} = \frac{\sqrt{25 * 5}}{2} = \frac{5\sqrt{5}}{2}$. Therefore, we have $x + \frac{11}{2} = ±\frac{5\sqrt{5}}{2}$. This step is critical because it allows us to eliminate the square and move closer to finding the values of x that satisfy the original equation. Remembering to consider both positive and negative roots is crucial for obtaining the complete solution set.

6. Solving for x

At this stage, our equation is $x + \frac{11}{2} = ±\frac{5\sqrt{5}}{2}$. To finally solve for x, we need to isolate it by subtracting $\frac{11}{2}$ from both sides of the equation. This gives us $x = -\frac{11}{2} ± \frac{5\sqrt{5}}{2}$. This equation represents two distinct solutions for x, one with the addition of $\frac{5\sqrt{5}}{2}$ and the other with its subtraction. These two solutions are the roots of the quadratic equation, representing the points where the parabola intersects the x-axis. By isolating x, we have successfully found the values that satisfy the original equation.

7. The Two Solutions

The equation $x = -\frac{11}{2} ± \frac{5\sqrt{5}}{2}$ provides us with two solutions for x. Let's explicitly write them out: The first solution is $x_1 = -\frac{11}{2} + \frac{5\sqrt{5}}{2}$ and the second solution is $x_2 = -\frac{11}{2} - \frac{5\sqrt{5}}{2}$. These are the two values of x that, when substituted back into the original equation, will make the equation true. In decimal approximations, these solutions are approximately $x_1 ≈ -0.9098$ and $x_2 ≈ -10.0902$. These solutions are the points where the graph of the quadratic equation $x^2 + 11x - 1 = 0$ intersects the x-axis. Understanding that quadratic equations can have two solutions, one solution, or no real solutions is fundamental in mathematics.

8. Verification (Optional)

To ensure the accuracy of our solutions, it is always a good practice to verify them. We can do this by substituting each value of x back into the original equation, $x^2 + 11x - 1 = 0$, and checking if the equation holds true. For $x_1 = -\frac{11}{2} + \frac{5\sqrt{5}}{2}$, substituting this value into the equation and simplifying should result in 0. Similarly, for $x_2 = -\frac{11}{2} - \frac{5\sqrt{5}}{2}$, the same process should yield 0. This verification step helps to catch any potential errors made during the solving process and reinforces the understanding of the solution. While optional, verification is a valuable tool in mathematical problem-solving.

9. Conclusion

In conclusion, we have successfully solved the quadratic equation $x^2 + 11x + \frac{121}{4} = \frac{125}{4}$ by employing the method of completing the square. We simplified the equation, completed the square, isolated the squared term, took the square root, and solved for x, arriving at the two solutions: $x_1 = -\frac{11}{2} + \frac{5\sqrt{5}}{2}$ and $x_2 = -\frac{11}{2} - \frac{5\sqrt{5}}{2}$. This step-by-step approach demonstrates a systematic way to tackle quadratic equations, which are prevalent in various mathematical and scientific contexts. Mastering these techniques not only enhances problem-solving skills but also provides a solid foundation for more advanced mathematical concepts. The ability to solve quadratic equations is a fundamental skill that opens doors to a deeper understanding of mathematics and its applications.