If a +√x= b then x =

Rationale

x = (b - a)²

To isolate x in the given equation a + √x = b, we first rearrange it to find that √x = b - a. Squaring both sides leads us directly to x = (b - a)², confirming that this expression accurately represents the value of x in terms of a and b.

A (√b-√a) YOUR ANSWER

This option incorrectly suggests that x can be expressed as the difference of the square roots of b and a. The original equation involves the square root of x, not the square roots of a and b separately. Therefore, this expression does not align with the algebraic manipulation required to isolate x.

B (√(b-1)) YOUR ANSWER

This choice misrepresents the relationship between a and b in the equation. The expression indicates that x is the square root of (b - 1), which does not derive from the manipulation of the equation a + √x = b. The correct operation does not involve subtracting 1 from b, making this choice invalid.

C ((b-a)²) CORRECT

This choice accurately represents the value of x after rearranging and squaring the equation. By isolating the square root on one side and then squaring both sides, we arrive at x = (b - a)², which is directly derived from the original equation, confirming its correctness.

D (b²-a²) YOUR ANSWER

This option incorrectly suggests that x can be represented as the difference of the squares of b and a. The operation described does not reflect the steps taken to isolate x from the equation a + √x = b. The difference of squares does not apply here, making this choice incorrect.

Conclusion

The correct manipulation of the equation a + √x = b leads us to isolate and express x as (b - a)². The other choices either misunderstand the operations required or misrepresent the relationship defined in the equation. Understanding these algebraic principles is crucial for accurately solving such problems in mathematics.