Решение №4525 Решите уравнение 1/(x-1)^2+3/(x-1)-10=0.

Решение:

\frac{1}{(x-1)^{2}}+\frac{3}{x-1}-10=0\\\frac{1\cdot 1+3\cdot (x-1)-10\cdot (x-1)^{2}}{(x-1)^{2}}=0\\\frac{1+3x-3-10\cdot (x^{2}-2x+1)}{(x-1)^{2}}=0\\\frac{1+3x-3-10x^{2}+20x-10}{(x-1)^{2}}=0\\\frac{-10x^{2}+23x-12}{(x-1)^{2}}=0\:{\color{Blue} |\cdot (x-1)^{2}}

ОДЗ: (х – 1)² ≠ 0
х – 1 ≠ 0
x ≠ 1

–10x² + 23x – 12 = 0
D = 23² – 4·(–10)·(–12) = 49 = 7²
x_{1}=\frac{-23+7}{2\cdot (-10)}=\frac{-16}{-20}=\frac{4}{5}\\x_{2}=\frac{-23-7}{2\cdot (-10)}=\frac{-30}{-20}=\frac{3}{2}

Ответ: \frac{4}{5};\frac{3}{2}.