Solving (x-10)^2 + 10 = 100 A Step-by-Step Guide
Introduction
In this article, we will delve into the step-by-step process of solving the algebraic equation (x-10)^2 + 10 = 100. This type of equation, a quadratic equation in disguise, often appears in mathematics courses and problem-solving scenarios. Understanding how to solve it not only strengthens your algebraic skills but also provides a foundation for tackling more complex mathematical problems. We will break down each step, providing clear explanations and justifications to ensure a thorough understanding of the solution. This comprehensive guide aims to equip you with the necessary tools and techniques to confidently approach similar equations in the future. Whether you are a student preparing for an exam or someone looking to brush up on your math skills, this article offers a valuable resource for mastering quadratic equations.
Understanding the Equation
Before diving into the solution, it's crucial to understand the structure of the equation (x-10)^2 + 10 = 100. This is a quadratic equation because the variable 'x' is raised to the power of 2. However, it's not in the standard form of a quadratic equation, which is ax^2 + bx + c = 0. Our equation is presented in a slightly modified form that requires us to perform some algebraic manipulations to isolate 'x'. The presence of the squared term (x-10)^2 indicates that we will likely have two solutions for 'x'. This is a fundamental characteristic of quadratic equations. Recognizing the structure and potential solutions is the first step towards successfully solving the equation. Understanding the components and their relationships will guide us through the appropriate steps to find the values of 'x' that satisfy the equation. The equation combines a squared term, a constant, and an equality, all of which play a role in the solution process. By carefully analyzing each element, we can develop a strategic approach to solve for 'x'.
Step-by-Step Solution
1. Isolate the Squared Term
The first step in solving the equation (x-10)^2 + 10 = 100 is to isolate the squared term, which is (x-10)^2. To do this, we need to remove the constant term (+10) from the left side of the equation. We achieve this by subtracting 10 from both sides of the equation. This maintains the equality and moves us closer to isolating the squared term. The equation now becomes:
(x-10)^2 + 10 - 10 = 100 - 10
This simplifies to:
(x-10)^2 = 90
Now, the squared term is isolated on one side of the equation, making it easier to proceed with the next steps. This isolation is crucial because it allows us to apply the square root property, which is a key technique for solving quadratic equations in this form. By isolating the squared term, we've effectively set the stage for undoing the square and finding the values of 'x'.
2. Take the Square Root of Both Sides
Now that we have isolated the squared term (x-10)^2 = 90, the next step is to take the square root of both sides of the equation. This operation is crucial because it undoes the square, allowing us to work with the expression inside the parentheses. However, it's essential to remember that when we take the square root of a number, we must consider both the positive and negative roots. This is because both the positive and negative values, when squared, will result in the same positive number. Therefore, taking the square root of both sides gives us:
√((x-10)^2) = ±√90
This simplifies to:
x - 10 = ±√90
The symbol '±' indicates that we have two possible solutions: one where the square root of 90 is positive, and another where it is negative. This is a fundamental aspect of solving quadratic equations, as they often have two distinct solutions.
3. Simplify the Square Root
Before we proceed further, it's beneficial to simplify the square root of 90 (√90). Simplifying radicals makes the equation easier to work with and provides a cleaner solution. To simplify √90, we look for perfect square factors within 90. We can express 90 as a product of its prime factors: 90 = 2 * 3^2 * 5. Notice that 3^2 = 9 is a perfect square factor. Therefore, we can rewrite √90 as:
√90 = √(9 * 10) = √9 * √10 = 3√10
Now, substituting this simplified form back into our equation, we have:
x - 10 = ±3√10
This simplification makes the equation more manageable and allows us to express the solutions in a more concise form. The simplified radical, 3√10, is easier to handle in subsequent calculations and provides a more accurate representation of the solutions.
4. Isolate x
To isolate 'x' and find its values, we need to eliminate the constant term (-10) on the left side of the equation x - 10 = ±3√10. We can achieve this by adding 10 to both sides of the equation. This maintains the equality and moves us closer to solving for 'x'. Adding 10 to both sides gives us:
x - 10 + 10 = 10 ± 3√10
This simplifies to:
x = 10 ± 3√10
Now, 'x' is isolated on one side of the equation, and we have two possible solutions, one with the positive square root and one with the negative square root. This reflects the quadratic nature of the equation, which typically results in two solutions.
5. Determine the Two Solutions
From the equation x = 10 ± 3√10, we can identify the two distinct solutions for 'x'. The '±' symbol indicates that we have two possibilities: one where we add 3√10 to 10, and another where we subtract 3√10 from 10. Therefore, the two solutions are:
x1 = 10 + 3√10
x2 = 10 - 3√10
These are the exact solutions for the equation (x-10)^2 + 10 = 100. If we need approximate decimal values for these solutions, we can use a calculator to evaluate the square root of 10 and perform the necessary arithmetic. However, leaving the solutions in this form, with the radical, is often preferred as it represents the exact values. These two solutions represent the points where the original equation holds true, and they are the culmination of our step-by-step solving process.
Approximate Solutions
While the exact solutions x1 = 10 + 3√10 and x2 = 10 - 3√10 provide a precise representation, it's often helpful to obtain approximate decimal values for practical applications or for a better understanding of the magnitude of the solutions. To find the approximate solutions, we can use a calculator to evaluate the square root of 10, which is approximately 3.162. Then, we substitute this value into our solutions:
For x1 = 10 + 3√10:
x1 ≈ 10 + 3 * 3.162
x1 ≈ 10 + 9.486
x1 ≈ 19.486
For x2 = 10 - 3√10:
x2 ≈ 10 - 3 * 3.162
x2 ≈ 10 - 9.486
x2 ≈ 0.514
Therefore, the approximate solutions for the equation (x-10)^2 + 10 = 100 are approximately 19.486 and 0.514. These values give us a clearer sense of where the solutions lie on the number line and can be useful for graphing or other applications where a decimal approximation is preferred.
Verification
To ensure the accuracy of our solutions, it's crucial to verify them by substituting them back into the original equation, (x-10)^2 + 10 = 100. This step confirms that the values we found for 'x' indeed satisfy the equation. Let's verify both solutions:
Verification of x1 = 10 + 3√10
Substitute x1 into the equation:
((10 + 3√10) - 10)^2 + 10 = 100
Simplify:
(3√10)^2 + 10 = 100
9 * 10 + 10 = 100
90 + 10 = 100
100 = 100
This confirms that x1 = 10 + 3√10 is a valid solution.
Verification of x2 = 10 - 3√10
Substitute x2 into the equation:
((10 - 3√10) - 10)^2 + 10 = 100
Simplify:
(-3√10)^2 + 10 = 100
9 * 10 + 10 = 100
90 + 10 = 100
100 = 100
This confirms that x2 = 10 - 3√10 is also a valid solution.
Since both solutions, x1 = 10 + 3√10 and x2 = 10 - 3√10, satisfy the original equation, we can confidently conclude that our solutions are correct. This verification step is a vital part of the problem-solving process, ensuring accuracy and reinforcing our understanding of the equation.
Conclusion
In this comprehensive guide, we have successfully solved the equation (x-10)^2 + 10 = 100 using a step-by-step approach. We began by understanding the structure of the equation, recognizing it as a quadratic equation in a non-standard form. We then systematically isolated the squared term, took the square root of both sides, simplified the radical, and isolated 'x' to find the two solutions: x1 = 10 + 3√10 and x2 = 10 - 3√10. We also determined the approximate decimal values for these solutions, which are approximately 19.486 and 0.514. Finally, we verified our solutions by substituting them back into the original equation, confirming their accuracy. This process not only demonstrates the solution to this specific equation but also provides a framework for solving similar quadratic equations. By mastering these techniques, you can confidently tackle a wide range of algebraic problems. Understanding the underlying principles and practicing these steps will enhance your mathematical skills and problem-solving abilities. Remember, the key to success in mathematics is a combination of understanding concepts, applying techniques, and verifying results.
