1. **State the problem:** Solve the quadratic equation $$3x^2 + 228 = 54x$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$3x^2 - 54x + 228 = 0$$ 3. **Simplify the equation:** Divide every term by 3 to simplify: $$\cancel{3}x^2 - \cancel{3}18x + \cancel{3}76 = 0 \implies x^2 - 18x + 76 = 0$$ 4. **Use the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=1$, $b=-18$, and $c=76$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-18)^2 - 4 \times 1 \times 76 = 324 - 304 = 20$$ 6. **Find the roots:** $$x = \frac{-(-18) \pm \sqrt{20}}{2 \times 1} = \frac{18 \pm \sqrt{20}}{2}$$ 7. **Simplify the square root:** $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ 8. **Final solutions:** $$x = \frac{18 \pm 2\sqrt{5}}{2} = 9 \pm \sqrt{5}$$ **Answer:** The solutions to the equation are $$x = 9 + \sqrt{5}$$ and $$x = 9 - \sqrt{5}$$.