Решите уравнение: [tex]125* 2^{4x}-9*20^{x+1} +64* 25^{x} =0 [/tex]. Created by welcome124. algebra-ru

l :Замена: или или Ответ:

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welcome124 @welcome124

July 2022 1 8 Report

Решите уравнение: [tex]125* 2^{4x}-9*20^{x+1} +64* 25^{x} =0 [/tex]

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mukus13

Verified answer

$125* 2^{4x} -9* 20^{x+1} +64* 5^{2x} =0$

$125* 4^{2x} -9*20* 20^{x} +64* 5^{2x} =0$

$125* 4^{2x} -180* 20^{x} +64* 5^{2x} =0$ l : $5^{2x}$

$125* (\frac{4}{5}) ^{2x} -180* ( \frac{4}{5} )^{x} +64 =0$

Замена: $( \frac{4}{5} )^{x}=t,$ $t\ \textgreater \ 0$

$125t^2-180t+64=0$

$D_1=(\frac{b}{2} )^2-ac=90^2-125*64=8100-8000=100$

$t_1= \frac{90+10}{125}= \frac{4}{5}$

$t_2= \frac{90-10}{125}= \frac{16}{25}$

$( \frac{4}{5}) ^{x} =\frac{4}{5}$ или $( \frac{4}{5}) ^{x} =\frac{16}{25}$

$x=1$ или $( \frac{4}{5}) ^{x} =(\frac{4}{5})^2$

$x=2$

Ответ: $1;$ $2$

7 votes Thanks 3

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