Evaluate (x - √x)²: Expanding a Square Root Expression

To solve the problem, we will apply the square of a difference formula to the expression $(x-\sqrt{x})^2$.

The formula for the square of a difference is:

• $(a-b)^2 = a^2 - 2ab + b^2$

For our expression $(x-\sqrt{x})^2$, let $a = x$ and $b = \sqrt{x}$.

Substituting into the formula, we have:

$(x-\sqrt{x})^2 = x^2 - 2x\sqrt{x} + (\sqrt{x})^2$

Now, calculate each term separately:

• The first term: $x^2$ remains as is.
• The second term: $-2x\sqrt{x}$ is already simplified.
• The third term: $(\sqrt{x})^2$ simplifies to $x$.

Substitute these back into the expanded expression:

$(x-\sqrt{x})^2 = x^2 - 2x\sqrt{x} + x$

Simplifying further:

$x^2 - 2x\sqrt{x} + x = x^2 + x - 2x\sqrt{x}$

To match the provided multiple-choice answers, factor common elements:

$x[x - 2\sqrt{x} + 1]$

Therefore, the simplified expression is:

$x[x-2\sqrt{x}+1]$

The correct answer choice, as compared to the provided options, is:

$x[x-2\sqrt{x}+1]$