Question

O valor de x € N, tal que

Answer 1

$\frac{(x + 1)! \cdot (3x + 2)!}{(3x + 1)! \cdot (x - 1)x!} = 24 \\ \frac{(x + 1) \cdot x! \cdot (3x + 2) \cdot (3x + 1)!}{(3x + 1)! \cdot (x - 1) \cdot x!} = 24 \\ \frac{(x + 1) \cdot (3x + 2)}{x - 1} = 24 \\ 3 {x}^{2} + 2x + 3x + 2 = 24x - 24 \\ 3 {x}^{2} + 5x - 24x + 2 + 24 = 0 \\ \\ 3 {x}^{2} - 19x + 26 = 0 \\ \\ \Delta = {b}^{2} - 4ac = {19}^{2} - 4 \cdot 3 \cdot 26 \\ \Delta = 361 - 312 \\ \Delta = 49 \\ \\ x_1 = \frac{ - b + \sqrt{ \Delta} }{2a} = \frac{19 + \sqrt{49} }{2 \cdot 3} = \frac{19 + 7}{6} = \frac{26}{6} \\ x_1 = \frac{13}{3} \\ x_2 = \frac{ - b - \sqrt{ \Delta} }{2a} = \frac{19 - \sqrt{49} }{2 \cdot 3} = \frac{19 - 7}{6} = \frac{12}{6} \\ x_2 = 2$

Como $x \in \mathbb{N}$ então:

$\boxed{\boxed{ \textrm{S = \{2\}}}}$