Find The Equation Equivalent To C When 10b = 5(√c + 2) And One Solution Is C = (2b - 2)^2.
In this comprehensive exploration, we will delve into the mathematical problem presented by the equation $10b = 5(\sqrt{c} + 2)$. Our primary goal is to understand how solving this equation for c leads to the solution $c = (2b - 2)^2$, and subsequently, to identify which of the provided options is an equivalent equation for finding c. This involves a step-by-step analysis of the given equation, algebraic manipulations, and a comparative examination of the answer choices. By the end of this article, readers will gain a clear understanding of the solution process and the underlying mathematical principles.
Understanding the Initial Equation
The journey begins with the given equation: $10b = 5(\sqrt{c} + 2)$. This equation relates b and c through a square root function. To isolate c, we need to methodically reverse the operations applied to it. The first step is to simplify the equation by dividing both sides by 5. This simplification makes the equation easier to handle and sets the stage for further algebraic manipulation. This step is crucial for isolating the term containing the square root of c, which is essential for ultimately solving for c. By carefully following each step, we can ensure an accurate and efficient solution to the problem.
Simplifying the Equation: A Detailed Approach
Dividing both sides of the equation $10b = 5(\sqrtc} + 2)$ by 5 is a fundamental algebraic step. This operation maintains the equality of the equation while simplifying its terms. The result of this division is + 2$. This simplified form is significantly easier to work with. Now, our focus shifts to isolating the square root term, which is a critical step in solving for c. This process involves further algebraic manipulations, ensuring that each step is mathematically sound and moves us closer to our goal. The simplified equation provides a clearer path towards finding the value of c.
Isolating the Square Root: The Next Critical Step
To further isolate c, we need to isolate the term $\sqrt{c}$. This can be achieved by subtracting 2 from both sides of the equation $2b = \sqrt{c} + 2$. Performing this subtraction yields the equation $2b - 2 = \sqrt{c}$. This step is crucial because it brings us closer to expressing c in terms of b. By isolating the square root term, we set the stage for eliminating the square root and directly solving for c. The equation $2b - 2 = \sqrt{c}$ represents a significant milestone in our solution process, paving the way for the final steps.
Solving for c: Squaring Both Sides
Now that we have the equation $2b - 2 = \sqrt{c}$, the next logical step to solve for c is to eliminate the square root. This is achieved by squaring both sides of the equation. Squaring both sides maintains the equality and effectively removes the square root, allowing us to express c directly in terms of b. This step is a pivotal moment in the solution process, as it transforms the equation into a form where c can be easily isolated and determined. By carefully executing this step, we move closer to identifying the equivalent equation for c.
The Result of Squaring: Expressing c in Terms of b
Squaring both sides of the equation $2b - 2 = \sqrt{c}$ gives us $(2b - 2)^2 = c$, which is the given solution for c. This equation explicitly expresses c as a function of b. The process of squaring both sides has successfully eliminated the square root, providing us with a direct relationship between c and b. Now, we have a clear understanding of how c is related to b, which is essential for identifying the equivalent equation from the given options. This result serves as the foundation for our subsequent analysis and comparison of the answer choices.
Identifying the Equivalent Equation
Our derived solution for c is $c = (2b - 2)^2$. To find the equivalent equation among the given options, we need to compare each option with this solution. This involves understanding the algebraic manipulations that can transform one equation into another. The goal is to identify which option, when simplified or rearranged, matches our derived solution. This comparative analysis is a crucial step in solving the problem, as it ensures that we select the correct equivalent equation for c. By carefully examining each option, we can determine which one accurately represents the relationship between c and b.
Expanding the Derived Solution: A Key to Comparison
Expanding the derived solution $c = (2b - 2)^2$ will provide us with a polynomial form that can be easily compared with the given options. Expanding the square, we get: $c = (2b - 2)(2b - 2) = 4b^2 - 8b + 4$. This expanded form is crucial for matching the solution with the answer choices, as it presents the equation in a standard polynomial format. Now, we have a clear expression for c in terms of b, which will facilitate the process of identifying the equivalent equation. The expanded form serves as a benchmark for our comparison, ensuring an accurate selection of the correct answer.
Analyzing the Answer Choices
Now, let's analyze the given answer choices to determine which one is equivalent to $c = 4b^2 - 8b + 4$. We need to carefully examine each option and perform any necessary algebraic manipulations to see if it matches our expanded solution. This process involves comparing the coefficients and terms in each option with those in our derived equation. By systematically evaluating each choice, we can accurately identify the one that represents the same relationship between c and b. This step-by-step analysis is critical for ensuring that we select the correct equivalent equation.
Option A: A Detailed Examination
Option A is given as $c = 10b - 10 - 5$, which simplifies to $c = 10b - 15$. This is a linear equation, while our derived solution $c = 4b^2 - 8b + 4$ is a quadratic equation. Therefore, Option A is not equivalent to our solution. The linear nature of Option A immediately distinguishes it from the quadratic relationship we established between c and b. This discrepancy allows us to quickly eliminate Option A from consideration. The comparison highlights the importance of matching the form of the equation, as well as the specific coefficients and terms.
Option B: A Thorough Comparison
Option B is given as $c = (10b - 10 - 5)^2$, which simplifies to $c = (10b - 15)^2$. Expanding this, we get $c = 100b^2 - 300b + 225$, which is clearly not equivalent to our solution $c = 4b^2 - 8b + 4$. The coefficients and terms in Option B's expanded form differ significantly from those in our derived solution. This disparity confirms that Option B does not represent the same relationship between c and b. The detailed comparison underscores the need for a precise match between the equations to ensure equivalence.
Correct Option: Confirming the Solution
The correct equivalent equation for finding c is $c = (2b - 2)^2$, which was our derived solution. This equation, when expanded, gives us $c = 4b^2 - 8b + 4$. This matches the result we obtained by squaring both sides of the simplified equation. Therefore, the correct answer is the equation that represents this relationship between c and b. The confirmation of the solution reinforces the accuracy of our step-by-step analysis and algebraic manipulations. This final validation ensures that we have successfully identified the equivalent equation for c.
Conclusion
In conclusion, by systematically simplifying the given equation $10b = 5(\sqrt{c} + 2)$, isolating the square root term, and squaring both sides, we arrived at the solution $c = (2b - 2)^2$. Through careful analysis and comparison, we identified this as the equivalent equation for finding c. This process highlights the importance of algebraic manipulation and step-by-step problem-solving in mathematics. By understanding the underlying principles and applying them methodically, we can successfully solve complex equations and identify equivalent forms. This comprehensive exploration demonstrates the power of algebraic techniques in unraveling mathematical relationships and finding solutions.