Solve the square root of (x squared - 6x + 1) equals 5 minus 2x.

Step-by-step solution

1. 1

   State the two conditions any solution must meet. The right side must be nonnegative, $5-2x\ge0$, i.e. $x\le\tfrac52$; and the radicand must be nonnegative, $x^2-6x+1\ge0$, i.e. $x\le 3-2\sqrt2$ or $x\ge 3+2\sqrt2$. Both are necessary, so candidates would have to satisfy $x \le 3-2\sqrt2 \approx 0.1716$.

2. 2

   Square both sides. $x^2-6x+1 = (5-2x)^2 = 25-20x+4x^2.$

3. 3

   Collect into standard form. Moving everything to one side: $0 = 25-20x+4x^2-x^2+6x-1 = 3x^2-14x+24.$

4. 4

   Compute the discriminant. $\Delta = (-14)^2-4(3)(24) = 196-288 = -92 < 0.$ A negative discriminant means the quadratic has no real roots at all.

5. 5

   Conclude. Since even the squared equation has no real solution, the original radical equation has no real solution. There is nothing left to test against the domain conditions.

6. 6

   Sanity-check with a value. At $x=0$ (which satisfies both domain conditions) the left side is $\sqrt{1}=1$ and the right side is $5$; as $x$ decreases the right side grows faster than the root, so the two curves never meet.