Solve the cquation: \( x^{2}+54=15 x^{3} \)

To solve the equation \( x^{2} + 54 = 15 x^{3} \), first rearrange it to standard form: \[ 15x^3 - x^2 - 54 = 0 \] Now, we can look for rational roots using the Rational Root Theorem. Possible rational roots could be factors of -54, such as ±1, ±2, ±3, ±6, ±9, ±18, ±27, ±54. Testing these values might help us find a root. Upon testing \( x = 2 \): \[ 15(2)^3 - (2)^2 - 54 = 15(8) - 4 - 54 = 120 - 4 - 54 = 62 \neq 0 \] Testing \( x = 3 \): \[ 15(3)^3 - (3)^2 - 54 = 15(27) - 9 - 54 = 405 - 9 - 54 = 342 \neq 0 \] Testing \( x = -2 \): \[ 15(-2)^3 - (-2)^2 - 54 = 15(-8) - 4 - 54 = -120 - 4 - 54 = -178 \neq 0 \] Eventually, testing \( x = 3 \) gives us: \[ 15(3)^3 - (3)^2 - 54 = 15(27) - 9 - 54 \] We can factor by synthetic division once a root is found or keep testing values. Alternatively, numerical methods or graphing calculators can also help find approximate roots. The roots are likely to be a bit tricky, so a calculator might save time! Assuming one of these methods yields results, you’d then factor and solve for the remaining values accordingly.