The Correct Answer and Explanation is:
One of the solutions to the given equation is x = sqrt(1521 + c^2). The other possible solution is x = -sqrt(1521 + c^2).
To find the solutions to the equation x^2 / sqrt(x^2 – c^2) = c^2 / sqrt(x^2 – c^2) + 39, we can begin by simplifying the expression. The first step involves consolidating the terms that contain the variable x and the constant c. Both fractions in the equation share a common denominator, which is sqrt(x^2 – c^2). This structure suggests that we can simplify the equation by moving the term c^2 / sqrt(x^2 – c^2) to the left side. We do this by subtracting it from both sides of the equation, which results in the following:
x^2 / sqrt(x^2 – c^2) – c^2 / sqrt(x^2 – c^2) = 39
Now that the terms with the common denominator are on the same side, we can combine them into a single fraction:
(x^2 – c^2) / sqrt(x^2 – c^2) = 39
The expression on the left side can be simplified using the rules of exponents. The numerator is x^2 – c^2, and the denominator is the square root of that same expression. An expression divided by its square root simplifies to the square root of that expression. For example, A / sqrt(A) = sqrt(A). Applying this rule, our equation becomes much simpler:
sqrt(x^2 – c^2) = 39
To isolate the variable x, we must first eliminate the square root. This is accomplished by squaring both sides of the equation:
(sqrt(x^2 – c^2))^2 = 39^2
x^2 – c^2 = 1521
Next, we add c^2 to both sides of the equation to isolate the x^2 term:
x^2 = 1521 + c^2
Finally, we solve for x by taking the square root of both sides. When taking a square root in this context, we must account for both the positive and negative roots:
x = ±sqrt(1521 + c^2)
This gives two distinct solutions for x. One solution is the positive root, x = sqrt(1521 + c^2), and the other is the negative root, x = -sqrt(1521 + c^2). The problem asks for one of the solutions, so either of these is a correct answer. It is also important to note that for the original equation to be defined, the term inside the square root, x^2 – c^2, must be greater than zero. Our solutions satisfy this condition, as x^2 = 1521 + c^2, which is inherently greater than c^2.
