Solve the square root of (x squared - 3) equals x minus 1.

Step-by-step solution

1. 1

   Impose the sign condition on the right-hand side. A square root output is never negative, so any solution must have $x-1 \ge 0 \quad\Longleftrightarrow\quad x \ge 1.$ This condition is what will later distinguish real solutions from extraneous ones.

2. 2

   Square both sides. $x^2-3 = (x-1)^2 = x^2-2x+1.$

3. 3

   Cancel the quadratic terms. $x^2$ appears on both sides and cancels, leaving a linear equation: $-3 = -2x+1.$ Whenever the leading terms match like this, the squared equation drops a degree.

4. 4

   Solve for x. $-4 = -2x \quad\Longrightarrow\quad x = 2.$

5. 5

   Test against both requirements. $x=2$ satisfies $x\ge1$, and the radicand is $2^2-3 = 1 \ge 0$. Substituting: $\sqrt{4-3} = \sqrt1 = 1$ and $2-1 = 1$. The two sides agree, so $x=2$ is a genuine solution.

6. 6

   Note why the check matters. Squaring can turn $\sqrt{A} = B$ into solutions of $\sqrt{A} = -B$; here only one candidate appeared and it survived, but in the sibling problem $\sqrt{x^2-6x+25} = 3-x$ the same procedure yields a contradiction instead.