Solving Cubic Equation: 9(x-1)³ = -125/27

To solve the equation 9(x-1)ﾳ = -125/27, we can start by simplifying the left side using the cube of a binomial formula:

(x-1)ﾳ = (x)ﾳ - 3(x)ﾲ(1) + 3(x)(1)ﾲ - 1ﾳ = xﾳ - 3xﾲ + 3x - 1

Substituting back into the original equation, we have:

9(xﾳ - 3xﾲ + 3x - 1) = -125/27

Expanding and simplifying:

9xﾳ - 27xﾲ + 27x - 9 = -125/27

Multiplying both sides by 27 to get rid of the fraction:

243xﾳ - 729xﾲ + 729x - 243 = -125

Adding 125 to both sides:

243xﾳ - 729xﾲ + 729x - 118 = 0

Dividing both sides by 9 to simplify:

27xﾳ - 81xﾲ + 81x - 13 = 0

We can use synthetic division or a factor theorem to check that x = 1/3 is a root of this equation. Dividing by (x - 1/3), we get:

27xﾲ - 54x + 39 = 0

This quadratic equation has no real solutions, so the only solution to the original equation is x = 1/3.