1. **State the problem:** Solve the quadratic equation $x^2 + 1 = 2x$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 + 1 - 2x = 0$$ which simplifies to $$x^2 - 2x + 1 = 0$$ 3. **Recognize the form:** The equation is a quadratic in standard form $ax^2 + bx + c = 0$ with $a=1$, $b=-2$, and $c=1$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Substitute the values: $$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \times 1 \times 1}}{2 \times 1} = \frac{2 \pm \sqrt{4 - 4}}{2}$$ 5. **Simplify under the square root:** $$\sqrt{4 - 4} = \sqrt{0} = 0$$ 6. **Calculate the roots:** $$x = \frac{2 \pm 0}{2} = \frac{2}{2} = 1$$ 7. **Interpretation:** There is one unique solution (a repeated root) at $x=1$. **Final answer:** $$x = 1$$