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1 vote
4. solve
e) 4^x-10.2^x+16=0
I think the dot means multiply?

User Kash
by
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2 Answers

3 votes
2^x=a
4^x=(2^2)^x=(2^x)^2=a^2

a^2-10a+16=0
Δ=10^2-4*1*16=100-64=36
√Δ=6
a1=(10+6)/2=16/2=8
a2=(10-6)/2=4/2=2
2^x=8=2^3⇒ x=3 or
2^x=2=2^1⇒ x=1


There are 2 solutions : x=1 or x=3
User Thomas Wouters
by
8.3k points
6 votes

4^x-10\cdot2^x+16=0\Rightarrow(2^2)^x-10\cdot2^x+16=0\Rightarrow(2^x)^2-10\cdot2^x+16=0


subtitute\ 2^x=t > 0\\\\t^2-10t+16=0\\t^2-2t-8t+16=0\\t(t-2)-8(t-2)=0\\(t-2)(t-8)=0\iff t-2=0\ or\ t-8=0\\\\t=2\ or\ t=8\\\\t=2^1\ or\ t=2^3\\\\therefore\ 2^x=2^1\ or\ 2^x=2^3\\\\Answer:\boxed{x=1\ or\ x=3}
User Luis Meraz
by
8.1k points

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