+0  
 
+3
613
4
avatar+253 

Solve the equation. 

e2x − 9ex + 8 = 0

e6x + 5e3x − 14 = 0

x 2 3 x −  4( 3 x) = 0

 

sally1  Jul 9, 2014

Best Answer 

 #3
avatar+20038 
+24

x 2 3 x −  4( 3 x) = 0    ?

$$\\x^2 3^x-4(3^x)=0 \\
3^x
(
x^2-4
)
=0 \\
3^x
(
x-2
)
(
x+2
)
=0\\\\
\underbrace{3^x}_{=0}
\times(
\underbrace{x-2}_{=0}
)
\times(
\underbrace{x+2}_{=0}
)
=0 \\\\
\boxed{3^x=0} \quad | \quad \ln{} \\\\
\ln{(3^x)} = \ln{(0)} \\
x\ln{(3)} = \ln{(0)} \\
x=
{
\ln{(0)}
\over \ln{(3)} }
} \quad | \quad \ln{(0)} \mbox{ no solution !}\\\\
\boxed{x-2=0} \quad \Rightarrow \quad \boxed{x=x_1=2}\\\\
\boxed{x+2=0} \quad \Rightarrow \quad \boxed{x=x_2=-2}$$

heureka  Jul 10, 2014
 #1
avatar+20038 
+24

e2x − 9ex + 8 = 0  ?

 $$\\e^{2x}=e^xe^x \quad | \quad \mbox{ set }\quad \boxed{z=e^x}\\
z^2 -9z +8=0\\
\underbrace{1}_{a=1}z^2 \underbrace{-9}_{b=-9}z \underbrace{+8}_{c=8} =0\\
\boxed{z_{1,2}=
{
-b\pm\sqrt{b^2-4ac}
\over
2a
}
}\\\\
z_{1,2}=
{ 9\pm\sqrt{81-4*1*8}
\over
2*1 } \\\\
z_{1,2}=
{ 9\pm\sqrt{81-32}
\over
2} \\\\
z_{1,2}=
{ 9\pm7\over
2} \\\\
\boxed{z_1= 8 \quad z_2 = 1} \\\\
e^x=z \quad | \quad \ln\\\\
\ln{(e^x)}=\ln{(z)}\\\\
x\ln{(e)}=\ln{(z)} \quad | \quad \ln{(e)}= 1 \quad !\\\\
\boxed{x=\ln{(z)}} \\\\$$

$$\\x=x_1=\ln{(8)}=2.07944154168\\
x=x_2=\ln{(1)}=0\\
\boxed{x_1=2.07944154168 \qquad x_2=0}$$

heureka  Jul 10, 2014
 #2
avatar+20038 
+24

e6x + 5e3x − 14 = 0 ?

$$\\e^{6x}=e^{3x}e^{3x} \quad | \quad \mbox{ set }\quad \boxed{z=e^{3x}}\\
z^2 +5z -14=0\\
\underbrace{1}_{a=1}z^2 \underbrace{+5}_{b=5}z \underbrace{-14}_{c=-14} =0\\
\boxed{z_{1,2}=
{
-b\pm\sqrt{b^2-4ac}
\over
2a
}
}\\\\
z_{1,2}=
{ -5\pm\sqrt{25-4*1*(-14)}
\over
2*1 } \\\\
z_{1,2}=
{ -5\pm\sqrt{25+56}
\over
2} \\\\
z_{1,2}=
{ -5\pm9\over
2} \\\\
\boxed{z_1= 2 \quad z_2 = -7} \\\\
e^{3x}=z \quad | \quad \ln\\\\
\ln{(e^{3x})}=\ln{(z)}\\\\
3x\ln{(e)}=\ln{(z)} \quad | \quad \ln{(e)}= 1 \quad !\\\\
\boxed{x={1\over 3}\ln{(z)}} \\\\$$

$$\\x=x_1={
\ln{(2)}\over 3}
}
=0.23104906019\\ x=x_2={
\ln{(-7)}\over 3}
}
= \mbox{no solution !}\\\\
\boxed{x=0.23104906019}$$

heureka  Jul 10, 2014
 #3
avatar+20038 
+24
Best Answer

x 2 3 x −  4( 3 x) = 0    ?

$$\\x^2 3^x-4(3^x)=0 \\
3^x
(
x^2-4
)
=0 \\
3^x
(
x-2
)
(
x+2
)
=0\\\\
\underbrace{3^x}_{=0}
\times(
\underbrace{x-2}_{=0}
)
\times(
\underbrace{x+2}_{=0}
)
=0 \\\\
\boxed{3^x=0} \quad | \quad \ln{} \\\\
\ln{(3^x)} = \ln{(0)} \\
x\ln{(3)} = \ln{(0)} \\
x=
{
\ln{(0)}
\over \ln{(3)} }
} \quad | \quad \ln{(0)} \mbox{ no solution !}\\\\
\boxed{x-2=0} \quad \Rightarrow \quad \boxed{x=x_1=2}\\\\
\boxed{x+2=0} \quad \Rightarrow \quad \boxed{x=x_2=-2}$$

heureka  Jul 10, 2014
 #4
avatar+93691 
+6

Really nice work Heureka!

I'd give you more points if I could.  

Melody  Jul 11, 2014

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