1. **State the problem:** Solve the quadratic equation $x^2 = 5x - 4$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 - 5x + 4 = 0$$ 3. **Identify the quadratic form:** The equation is in standard form $ax^2 + bx + c = 0$ where $a=1$, $b=-5$, and $c=4$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times 4 = 25 - 16 = 9$$ 6. **Find the roots:** $$x = \frac{-(-5) \pm \sqrt{9}}{2 \times 1} = \frac{5 \pm 3}{2}$$ 7. **Evaluate each root:** - For $+$ sign: $x = \frac{5 + 3}{2} = \frac{8}{2} = 4$ - For $-$ sign: $x = \frac{5 - 3}{2} = \frac{2}{2} = 1$ **Final answer:** The solutions to the equation are $x = 4$ and $x = 1$.