Question

40БАЛОВ.Решите уравнение
6/x²-36 + 6/(x+6)² + 1/2x+12 =0

Answer 1

$\displaystyle\bf\\\frac{6}{x^{2} -36} +\frac{6}{(x+6)^{2} } +\frac{1}{2x+12} =0\\\\\\\frac{6}{(x+6)(x-6)}+\frac{6}{(x+6)^{2} } +\frac{1}{2(x+6)}=0\\\\\\\frac{6\cdot2(x+6)+6\cdot2(x-6)+(x+6)(x-6)}{2(x-6)(x+6)^{2} } =0\\\\\\\frac{12x+72+12x-72+x^{2} -36}{2(x-6)(x+6)^{2} }=0\\\\\\\frac{x^{2} +24x-36}{2(x-6)(x+6)^{2} } =0\\\\\\\left[\begin{array}{ccc}x^{2} +24x-36=0\\x-6\neq 0\\x+6\neq 0\end{array}\right\\\\\\\left[\begin{array}{ccc}x^{2} +24x-36=0\\x\neq 6\\x\neq -6\end{array}\right$

$\displaystyle\bf\\x^{2} +24x-36=0\\\\D=24^{2} -4\cdot(-36)=576+144=720=(12\sqrt{5} )^{2}\\\\\\x_{1} =\frac{-24-12\sqrt{5} }{2} =-12-6\sqrt{5} \\\\\\x_{2} =\frac{-24+12\sqrt{5} }{2} =6\sqrt{5}-12$