Solve the square root of (x squared - 6x + 25) equals 5.

Step-by-step solution

1. 1

   Check that squaring is legitimate. The right-hand side is the constant $5>0$, so no extra condition on $x$ is created; also $x^2-6x+25 = (x-3)^2+16 \ge 16 > 0$, so the radicand is always defined. Squaring here neither loses nor invents solutions.

2. 2

   Square both sides. $x^2-6x+25 = 25.$

3. 3

   Cancel and factor. The $25$ on both sides cancels: $x^2-6x = 0 \quad\Longrightarrow\quad x(x-6) = 0.$ Factoring out $x$ is faster and safer than the quadratic formula here.

4. 4

   Read off both roots. $x = 0 \quad\text{or}\quad x = 6.$

5. 5

   Verify each root in the original equation. At $x=0$: $\sqrt{0-0+25} = \sqrt{25} = 5$. At $x=6$: $\sqrt{36-36+25} = \sqrt{25} = 5$. Both check out - no extraneous roots.

6. 6

   See the symmetry. Completing the square gives $\sqrt{(x-3)^2+16} = 5$, i.e. $(x-3)^2 = 9$, so $x-3=\pm3$. The two roots are symmetric about the vertex $x=3$, which is why they are $0$ and $6$.