Question

3/x-5+8/x=2 решите пожалуйста ​

Answer 1

Решение и ответ:

$\displaystyle \frac{3}{{x-5}}+\frac{8}{x}=2$

$\displaystyle \frac{3}{{x-5}}+\frac{8}{x}-2=0$

$\displaystyle \frac{{3x}}{{x\left({x-5}\right)}}+\frac{{8(x-5)}}{{x\left({x-5}\right)}}-\frac{{2x(x-5)}}{{x\left({x-5}\right)}}=0$

$\displaystyle \frac{{3x+8(x-5)-2x(x-5)}}{{x\left({x-5}\right)}}=0$

$\displaystyle \frac{{3x+8x-40-2{x^2}+10x}}{{x\left({x-5}\right)}}=0$

$\displaystyle \frac{{-2{x^2}+21x -40}}{{x\left({x-5}\right)}}=0$

ОДЗ:

$\displaystyle x\left({x-5}\right) \ne 0$

$\displaystyle x \ne 0$

$\displaystyle x-5 \ne 0$

$\displaystyle x \ne 5$

Решим квадратное уравнение:

$\displaystyle -2{x^2}+21x -40=0$

$\displaystyle D={b^2}-4ac={21^2}-4\cdot(-2)\cdot(-40)=441-320=121$

$\displaystyle x=\frac{{-b\pm\sqrt D}}{{2a}}=\frac{{-21\pm\sqrt {121}}}{{2\cdot(-2)}}=\frac{{-21\pm11}}{{-4}}$

$\displaystyle {x_1}=\frac{{-21+11}}{{-4}}=\frac{{-10}}{{-4}}=\boxed{2.5}$

$\displaystyle {x_2}=\frac{{-21-11}}{{-4}}=\frac{{-32}}{{-4}}=\boxed{8}$