Rationale
x = (b - a)²
This equation is derived from rearranging the initial equation a + √x = b, leading to the conclusion that x can be expressed as the square of the difference between b and a. This transformation provides the correct algebraic expression for x in terms of a and b.
A) The expression √b - √a does not correctly represent x because it does not account for the relationships defined in the original equation. Instead, it incorrectly simplifies the square root operations, leading to an invalid result that cannot be derived from a + √x = b.
B) The expression √(b - 1) fails to connect with the original equation as it disregards both a and the square root of x. This choice does not maintain the integrity of the equation, as it does not reflect the necessary operations to isolate and express x accurately.
C) The expression (b - a)² correctly represents the value of x by isolating √x in the equation a + √x = b. This leads to the conclusion that √x = b - a, and squaring both sides gives x = (b - a)², substantiating it as the accurate solution.
D) The expression b² - a² represents a difference of squares, which is unrelated to the original equation. This choice does not follow logically from the operations performed on the equation a + √x = b and thus cannot be considered a valid expression for x.
Conclusion
In solving the equation a + √x = b, the correct solution is x = (b - a)², derived from isolating the square root term and squaring both sides. Other choices either misinterpret the operations necessary to isolate x or introduce unrelated mathematical concepts, underscoring the importance of careful algebraic manipulation.