Решите уравнения1)4^(2x) - 7 * 4^x + 16 = 02)2^(2x + 1) + 2^(x + 2) - 16 = 03)2*4^2x + 8 = 174)2 * 16^x - 7 * 4^x - 4 = 05)4^x + 6 ______ = 4 4^(1/2 - x/2)6)2^2x - 6 * 2^x + 8 = 07)4^x - 5 *...
Решите уравнения
1)4^(2x) - 7 * 4^x + 16 = 0
2)2^(2x + 1) + 2^(x + 2) - 16 = 0
3)2*4^2x + 8 = 17
4)2 * 16^x - 7 * 4^x - 4 = 0
5)4^x + 6
______ = 4
4^(1/2 - x/2)
6)2^2x - 6 * 2^x + 8 = 0
7)4^x - 5 * 2^x + 4 = 0
8)9^(x+1) + 3^(x+2) - 18 = 0
9)25^x - 24 * 5^(x-1) - 5^(log5 3) + 2 = 0
10)7^(x-2) + 38 * 3^x = 7^(x+1)
Гость
Ответ(ы) на вопрос:
Гость1) [latex] 4^{2x} -7* 4^{x}+16=0 [/latex]
[latex] 4^{x}=t;t>0 [/latex]
t² - 7t + 16 = 0
D= 49-64 <0
корней нет.
2) [latex]2* 2^{2x}+4* 2^{x}-16=0 [/latex]
[latex] 2^{x}=t;t>0 [/latex]
2t² + 4t - 16=0
t² + 2t - 8 = 0
D₁ = 1+8=9
t₁ = 2
t₂ = -4 не удов. усл. t>0
[latex] 2^{x}=2 [/latex]
x=1
3) [latex]2* 4^{2x}+8 = 17 [/latex]
[latex]2* 4^{2x}=9 [/latex]
[latex] 4^{2x}=4.5 [/latex]
[latex] 4^{2x}= 4^{ log_{4}4.5 } [/latex]
2x= log₄4.5
x = log₁₆4.5
4) [latex]2* 16^{x}-7* 4^{x}-4=0 [/latex]
[latex]2*4^{2x}-7* 4^{x}-4=0 [/latex]
[latex] 4^{x}=t; t>0 [/latex]
2t² - 7t - 4=0
D = 49+32=81
t₁ = 4
t₂ = -1/2 не удов. усл. t>0
[latex] 4^{x}=4 [/latex]
x=1
5) [latex] 4^{x}+6* 4^{ \frac{x}{2}- \frac{1}{2} } = 4 [/latex]
[latex] 4^{x}+ \frac{6* 4^{ \frac{x}{2} } }{ 4^{ \frac{1}{2} } }=4 [/latex]
[latex] 4^{x}+3* 4^{ \frac{x}{2} }=4 [/latex]
[latex] 4^{ \frac{x}{2} } =t; t>0[/latex]
t² + 3t = 4
t² + 3t - 4 = 0
t₁ = 1
t₂ = -4 не удов. усл.t>0
[latex] 4^{ \frac{x}{2} } =1[/latex]
[latex] \frac{x}{2} =0[/latex]
x=0
6) [latex] 2^{x}=t; t>0 [/latex]
t² - 6t + 8 = 0
D₁ = 9-8 = 1
t₁ = 4
t₂ = 2
[latex] 2^{x}=4 [/latex]
x=2
[latex] 2^{x}=2 [/latex]
x=1
7) [latex] 2^{2x}-5* 2^{x}+4=0 [/latex]
[latex] 2^{x}=t; t>0 [/latex]
t² - 5t + 4= 0
t₁ = 1
t₂ = 4
[latex] 2^{x}=1 [/latex]
x=0
[latex] 2^{x}=4 [/latex]
x=2
8) [latex] 9^{x+1}+ 3^{x+2}-18=0 [/latex]
[latex] 3^{2x}*9 + 9* 3^{x} -18=0 [/latex]
[latex] 3^{x}=t; t>0 [/latex]
9t² + 9t - 18 = 0
t₁ = 1
t₁ = -2 не удов.
[latex] 3^{x}=1 [/latex]
x= 0
9) [latex] 25^{x}-24* 5^{x-1}-3+2=0 [/latex]
[latex] 5^{2x}- \frac{24* 5^{x} }{5} -1=0[/latex]
5*[latex] 5^{2x}-24* 5^{x}-5 = 0 [/latex]
[latex] 5^{x}=t; t>0 [/latex]
5t² - 24t - 5 =0
D₁ = 144+25 = 169
t₁ = 5
t₂ = -1/5 не удов.
[latex] 5^{x}=5 [/latex]
x= 1
10) [latex] 7^{x-2}+38* 3^{x}= 7^{x+1} [/latex]
[latex] \frac{ 7^{x} }{49} +38* 3^{x}= 7^{x}*7 [/latex] I:[latex] 3^{x} [/latex]
[latex] \frac{1}{49}* ( \frac{7}{3}) ^{x}+38 = ( \frac{7}{3}) ^{x}*7 [/latex]
[latex]7* (\frac{7}{3}) ^{x} - \frac{1}{49}* ( \frac{7}{3}) ^{x}= 38 [/latex]
[latex] \frac{342}{49}* ( \frac{7}{3}) ^{x} = 38 [/latex]
[latex] ( \frac{7}{3}) ^{x}= \frac{49}{9} [/latex]
x = 2
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