+0  
 
+3
337
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+18715 
+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
Sort: 

4+0 Answers

 #1
avatar+18715 
+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+18715 
+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+18715 
+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+91045 
+6

Really nice work Heureka!

I'd give you more points if I could.  

Melody  Jul 11, 2014

8 Online Users

avatar
avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details