Question 11

x = (b - a)²

To isolate x from the equation a + √x = b, we first rearrange it to √x = b - a. Squaring both sides yields x = (b - a)².

A) √b - √a

This option suggests that x equals the difference between the square roots of b and a. However, this does not follow from the initial equation and fails to account for the squaring necessary to isolate x, making it incorrect.

B) √(b - 1)

This option implies that x is the square root of (b - 1). This does not comply with the transformation from the original equation since it disregards the relationship established between a, b, and √x, thus rendering it incorrect.

C) (b - a)²

This option is correct. By rearranging the equation a + √x = b to isolate √x, we find √x = b - a, and squaring both sides results in x = (b - a)², which accurately represents the solution.

D) b² - a²

This option represents the difference of squares, which does not relate to the transformation of the original equation. It does not follow the steps taken to isolate x from the equation, making it an incorrect choice.

Conclusion

The correct answer, x = (b - a)², directly follows from appropriately manipulating the equation a + √x = b. Each incorrect option either misrepresents the relationship between the variables or fails to apply the necessary mathematical operations, confirming that C is the only valid solution.