Mathematico | Mathematico

The solutions of the equation

(x + 3)^{3} - (x + 2)^{3} = 61

are $α$ and $β$ , where $α < β$ .

Find the value of

α^{4} - β^{3}

Hint

You need to expand. Recall that

\begin{aligned} (x + 3)^{3} & = (x + 3) (x + 3) (x + 3) \\ = (x^{2} + 6 x + 9) (x + 3) \\ = x^{3} + 6 x^{2} + 9 x + 3 x^{2} + 18 x + 27 \end{aligned}

Do similar for $(x + 2)^{3}$ , simplify the whole equation with $0$ on the right hand side, and solve.

Solution

The hint gives us

\begin{aligned} (x + 3)^{3} & = (x + 3) (x + 3) (x + 3) \\ = (x^{2} + 6 x + 9) (x + 3) \\ = x^{3} + 6 x^{2} + 9 x + 3 x^{2} + 18 x + 27 \\ = x^{3} + 9 x^{2} + 27 x + 27 \end{aligned}

Very similarly, we have

(x + 2)^{3} = x^{3} + 6 x^{2} + 12 x + 8

So our equation becomes

\begin{aligned} (x^{3} + 9 x^{2} + 27 x + 27) - (x^{3} + 6 x^{2} + 12 x + 8) & = 61 \\ 3 x^{2} + 15 x + 19 & = 61 \\ 3 x^{2} + 15 x - 42 & = 0 \\ x^{2} + 5 x - 14 & = 0 \\ (x + 7) (x - 2) & = 0 \\ x & \in {- 7, 2} \end{aligned}