Find the solutions of equation 64(9^x) - 84 (12^x)+ 27(16^x) = 0.
Answers
Answered by
6
Answer: X= 1, 2
Step-by-step explanation:
Anand230703:
pls provide step by step explanation
Let’s start with a sub, we see that 9, 12, and 16 share in common a 3 and a 4, so…
a=3x
b=4x
This is probably the hardest part of the problem, determining which things to substitute. Now we plug it in:
64(32)x−84(3x4x)+27(42)x=0
64(3x)2−84(3x4x)+27(4x)2=0
64a2−84ab+27b2=0
Now we can factor this, (I just some brute force and got it on my second try)
(16a−9b)(4a−3b)=0
16a=9b
4a=3b
Now let’s resubstitute:
16(3x)=9(4x)
4(3x)=3(4x)
43=(43)x;x=1
169=(43)x;x=2
Wasn’t that just beautiful? I love algebra,
x=1,2
a=3x
b=4x
This is probably the hardest part of the problem, determining which things to substitute. Now we plug it in:
64(32)x−84(3x4x)+27(42)x=0
64(3x)2−84(3x4x)+27(4x)2=0
64a2−84ab+27b2=0
Now we can factor this, (I just some brute force and got it on my second try)
(16a−9b)(4a−3b)=0
16a=9b
4a=3b
Now let’s resubstitute:
16(3x)=9(4x)
4(3x)=3(4x)
43=(43)x;x=1
169=(43)x;x=2
Wasn’t that just beautiful? I love algebra,
x=1,2
can i have another question
Answered by
2
Hence 9×16=(12)
2
then we divide its both side by 12
x
and obtain
64.(
4
3
)
x
−84+27.(
3
4
)
x
=0
Let (
4
3
)
x
=t then equation (1) reduce in the form
64t
2
−84t+27=0
where t
1
=
4
3
andt
2
=
16
9
then
(
4
3
)
x
=(
4
3
)
1
and(
4
3
)
x
=(
4
3
)
2
∴x
1
=1andx
2
=2
Hence sum of roots is 3.
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