Question

resolva a equação:

$\binom{x}{2} = 15$

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Answer 1

$\large\boxed{\begin{array}{l} \rm{ \binom{x}{2} = 15 } \\ \\ \rm \dfrac{x!}{2! \cdot(x - 2)!} = 15 \\ \\ \rm \dfrac{(x - 2)! \cdot(x - 1) \cdot{x}}{2! \cdot(x - 2)! } = 15 \\ \\ \rm \dfrac{(x - 1) \cdot{x}}{2} = 15 \\ \\ \rm{x {}^{2} - x = 30} \\ \\ \rm{x {}^{2} - x - 30 = 0} \\ \\ \Delta = ( - 1) {}^{2} - 4 \cdot 1 \cdot (- 30) \\ \Delta = 1 + 120 \\ \Delta = 121 \\ \sqrt{\Delta} = \sqrt{121} = 11 \\ \\ \rm {x _1 = \dfrac{ - ( - 1) - 11}{2 \cdot1} } = \dfrac{ - 10}{2} = - 5 < 0 \\ \\ \rm{x_2 = \dfrac{ - ( - 1) + 11}{ 2 \cdot1} = \dfrac{ 12}{2} = 6 }\end{array}}$