Solving (x-5)^2 = 16 - 4x A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of finding the roots of a specific equation: extbf{(x-5)^2 = 16 - 4x}. We will explore the steps involved in solving this equation, providing a clear and comprehensive guide for readers. Understanding how to solve such equations is crucial for various mathematical applications and problem-solving scenarios. This detailed exploration will not only provide the solution but also enhance your understanding of algebraic manipulations and equation-solving techniques.
Understanding the Equation
Our journey begins with a clear understanding of the equation at hand. The equation extbf{(x-5)^2 = 16 - 4x} is a quadratic equation disguised in a slightly different form. To effectively solve it, we need to transform it into the standard quadratic form, which is ax^2 + bx + c = 0. This form allows us to easily identify the coefficients and apply various methods, such as factoring, completing the square, or using the quadratic formula, to find the roots. The roots of an equation are the values of the variable (in this case, x) that make the equation true. Identifying these roots is the ultimate goal of our problem-solving process.
Before we dive into the solution, let's break down the equation into its components. The left side, extbf{(x-5)^2}, represents the square of the binomial (x-5). This means we need to expand this expression using the distributive property or the FOIL method (First, Outer, Inner, Last). The right side, 16 - 4x, is a linear expression. Our objective is to manipulate both sides of the equation to bring it into the standard quadratic form. This involves expanding the squared term, combining like terms, and rearranging the equation to set it equal to zero. Understanding these initial steps is crucial for a successful solution.
Step-by-Step Solution
Now, let's embark on the step-by-step solution to find the roots of the equation. This process involves algebraic manipulation, which requires a solid understanding of mathematical principles. We will break down each step in detail, ensuring that the logic behind each operation is clear.
1. Expanding the Left Side
The first step is to expand the left side of the equation, extbf{(x-5)^2}. This means multiplying the binomial (x-5) by itself. Using the FOIL method or the distributive property, we get:
(x-5)^2 = (x-5)(x-5) = x^2 - 5x - 5x + 25 = x^2 - 10x + 25
2. Rewriting the Equation
Now that we've expanded the left side, we can rewrite the original equation as:
x^2 - 10x + 25 = 16 - 4x
3. Rearranging to Standard Quadratic Form
To bring the equation into the standard quadratic form (ax^2 + bx + c = 0), we need to move all terms to one side of the equation. Let's add 4x to both sides and subtract 16 from both sides:
x^2 - 10x + 25 + 4x - 16 = 0
4. Combining Like Terms
Next, we combine like terms to simplify the equation:
x^2 - 6x + 9 = 0
Now, we have the equation in the standard quadratic form, where a = 1, b = -6, and c = 9.
5. Factoring the Quadratic Equation
Now that we have the quadratic equation in standard form, we can solve it. One method is factoring. We need to find two numbers that multiply to 9 and add up to -6. These numbers are -3 and -3. Therefore, we can factor the equation as:
(x - 3)(x - 3) = 0
This can also be written as:
(x - 3)^2 = 0
6. Solving for x
To find the roots, we set each factor equal to zero:
x - 3 = 0
Solving for x, we get:
x = 3
Since both factors are the same, we have a repeated root.
Alternative Method: Quadratic Formula
While we successfully solved the equation by factoring, it's beneficial to know other methods. The quadratic formula is a universal method for solving quadratic equations, even when factoring is difficult or impossible. The quadratic formula is:
x = [-b ± √(b^2 - 4ac)] / (2a)
For our equation, x^2 - 6x + 9 = 0, we have a = 1, b = -6, and c = 9. Plugging these values into the quadratic formula, we get:
x = [6 ± √((-6)^2 - 4 * 1 * 9)] / (2 * 1)
x = [6 ± √(36 - 36)] / 2
x = [6 ± √0] / 2
x = 6 / 2
x = 3
As we can see, the quadratic formula confirms our solution: x = 3.
Verification of the Solution
It's always a good practice to verify our solution by plugging it back into the original equation. Let's substitute x = 3 into the equation (x-5)^2 = 16 - 4x:
(3-5)^2 = 16 - 4(3)
(-2)^2 = 16 - 12
4 = 4
Since the equation holds true, our solution x = 3 is correct.
Conclusion
In conclusion, the root of the equation extbf{(x-5)^2 = 16 - 4x} is x = 3. We arrived at this solution by expanding, rearranging, factoring, and verifying our answer. We also demonstrated the use of the quadratic formula as an alternative method. Mastering these techniques is essential for solving various algebraic problems. The key to success lies in understanding the underlying principles and practicing consistently. This step-by-step guide provides a solid foundation for tackling similar equations and enhancing your problem-solving skills in mathematics.
