Mathematico | Mathematico

The solutions of

(x + 5)^{3} - (x - 1)^{3} = 182

are $α$ and $β$ .

Find the value of

81 α β

Hint

Expand and simplify!

Solution

We begin by expanding and simplifying

\begin{aligned} (x + 5)^{3} - (x - 1)^{3} & = 182 \\ (x^{3} + 15 x^{2} + 75 x + 125) - (x^{3} - 3 x^{2} + 3 x - 1) & = 182 \\ 18 x^{2} + 72 x - 56 & = 0 \\ 9 x^{2} + 36 x - 28 & = 0 \\ (3 x + 14) (3 x - 2) & = 0 \\ x & \in {- \frac{14}{3}, \frac{2}{3}} \end{aligned}