Solve the square root of (x squared - 6x + 25) equals 3 minus x.

Step-by-step solution

1. 1

   Record the domain condition. A square root is never negative, so any solution must satisfy $3-x \ge 0 \quad\Longleftrightarrow\quad x \le 3.$ Skipping this step is what lets extraneous roots survive in problems of this type.

2. 2

   Square both sides. $x^2-6x+25 = (3-x)^2 = 9-6x+x^2.$

3. 3

   Cancel the matching terms. Both $x^2$ and $-6x$ appear on each side and cancel, leaving $25 = 9.$

4. 4

   Interpret the contradiction. No value of $x$ can make $25$ equal $9$, so the squared equation - and therefore the original one - has no solution.

5. 5

   Confirm geometrically. $\sqrt{x^2-6x+25} = \sqrt{(x-3)^2+16}$ is the distance from $(x,0)$ to $(3,4)$, which is always at least $4$; meanwhile $3-x$ would have to equal it, and comparing $\sqrt{(x-3)^2+16}$ with $3-x$ gives $(x-3)^2+16 > (3-x)^2$ for every $x$. The left side is strictly larger, always.

6. 6

   Contrast with the sibling problem. Replacing the right side by the constant $5$ gives $\sqrt{x^2-6x+25}=5$, which does have solutions $x=0$ and $x=6$. It is the variable right-hand side that creates the contradiction here.