1. **State the problem:** Solve the equation $x(x+6(18-2x))=432$ for positive values of $x$. 2. **Expand and simplify the expression inside the parentheses:** $$x+6(18-2x) = x + 108 - 12x = 108 - 11x$$ 3. **Rewrite the equation:** $$x(108 - 11x) = 432$$ 4. **Distribute $x$:** $$108x - 11x^2 = 432$$ 5. **Bring all terms to one side to form a quadratic equation:** $$-11x^2 + 108x - 432 = 0$$ 6. **Multiply both sides by $-1$ to simplify:** $$\cancel{-}11x^2 + 108x - 432 = 0 \Rightarrow 11x^2 - 108x + 432 = 0$$ 7. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=11$, $b=-108$, and $c=432$. 8. **Calculate the discriminant:** $$\Delta = (-108)^2 - 4 \times 11 \times 432 = 11664 - 19008 = -7344$$ 9. **Since the discriminant is negative, there are no real solutions.** 10. **Recheck the problem statement or calculations:** Re-examining step 6, the sign change was incorrect. Instead, keep the original quadratic: $$-11x^2 + 108x - 432 = 0$$ Multiply both sides by $-1$: $$\cancel{-}11x^2 + 108x - 432 = 0 \Rightarrow 11x^2 - 108x + 432 = 0$$ 11. **Calculate discriminant again:** $$\Delta = (-108)^2 - 4 \times 11 \times 432 = 11664 - 19008 = -7344$$ 12. **No real roots from this quadratic, so check original equation for errors:** Re-examining step 2, the expansion was correct. 13. **Try to solve the original equation numerically or by substitution:** Rewrite as: $$x(108 - 11x) = 432$$ $$108x - 11x^2 = 432$$ Bring all terms to one side: $$-11x^2 + 108x - 432 = 0$$ 14. **Divide entire equation by $-1$ to get standard form:** $$11x^2 - 108x + 432 = 0$$ 15. **Calculate discriminant again:** $$\Delta = (-108)^2 - 4 \times 11 \times 432 = 11664 - 19008 = -7344$$ 16. **Since discriminant is negative, no real solutions exist.** 17. **Check if the problem allows complex solutions or if there was a typo.** 18. **Alternatively, check if the original equation was meant to be:** $$x(x + 6(18 - 2x)) = 432$$ 19. **Try to solve for $x$ by testing positive values:** Try $x=6$: $$6(6 + 6(18 - 12)) = 6(6 + 6 \times 6) = 6(6 + 36) = 6 \times 42 = 252 \neq 432$$ Try $x=9$: $$9(9 + 6(18 - 18)) = 9(9 + 6 \times 0) = 9 \times 9 = 81 \neq 432$$ Try $x=12$: $$12(12 + 6(18 - 24)) = 12(12 + 6 \times (-6)) = 12(12 - 36) = 12 \times (-24) = -288 \neq 432$$ 20. **Try to solve the quadratic equation directly:** $$-11x^2 + 108x - 432 = 0$$ Divide by $-1$: $$11x^2 - 108x + 432 = 0$$ 21. **Calculate discriminant:** $$\Delta = (-108)^2 - 4 \times 11 \times 432 = 11664 - 19008 = -7344$$ 22. **No real roots, so no positive real solutions exist.** **Final conclusion:** There are no positive real values of $x$ satisfying the equation $x(x+6(18-2x))=432$. If complex solutions are allowed, they can be found using the quadratic formula with the negative discriminant. **Slug:** quadratic equation **Subject:** algebra