To solve the equation \( 2f^5 - 85 + 2f = 8 \), we first rearrange it to isolate terms involving \( f \).
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Start with the original equation: \[ 2f^5 - 85 + 2f = 8 \]
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Add 85 to both sides: \[ 2f^5 + 2f = 8 + 85 \] This simplifies to: \[ 2f^5 + 2f = 93 \]
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Now, we can factor out the 2 on the left side: \[ 2(f^5 + f) = 93 \]
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Divide both sides by 2: \[ f^5 + f = \frac{93}{2} \]
Now, let's check if any of the provided options could be a solution.
The potential solutions to check are:
- \( f = 12 \)
- \( f = \frac{8}{3} \)
- \( f = \frac{16}{6} \) (which simplifies to \( \frac{8}{3} \))
- \( f = 4 \)
We'll test these values one by one in the equation \( f^5 + f = \frac{93}{2} \).
For \( f = 12 \): \[ f^5 + f = 12^5 + 12 \] Calculating \( 12^5 \) will yield a very large number, thus we can immediately infer that \( 12^5 + 12 \) will be significantly greater than \( \frac{93}{2} \).
For \( f = \frac{8}{3} \): \[ \left(\frac{8}{3}\right)^5 + \frac{8}{3} \] Calculating \( \left(\frac{8}{3}\right)^5 \) yields: \[ \left(\frac{8}{3}\right)^5 = \frac{32768}{243} \] So the left side is: \[ \frac{32768}{243} + \frac{8}{3} = \frac{32768}{243} + \frac{648}{243} = \frac{33416}{243} \] This does not equal \( \frac{93}{2} \).
For \( f = \frac{16}{6} \) (or \( \frac{8}{3} \)): This has already been computed above.
For \( f = 4 \): \[ f^5 + f = 4^5 + 4 = 1024 + 4 = 1028 \] This is also much greater than \( \frac{93}{2} \).
None of these computed potential solutions yield the actual solution as \( f^5 + f = \frac{93}{2} \) is more of a challenging equation to attempt typical resolution methods without a calculator to approximate or graph the solution.
Given typical expectation if largely left ambiguous, the best-guess evaluation leads to a realization that none of the potential common answers match such conditions.
For accurate evaluation, a numerical method or root solving techniques would be suggested to provide the possible value range within the specific conditions noted.
Based on direct checking, the closest theoretical solution would lead toward simpler evaluation or using software to sketch for precise roots. If linear evaluation needed, it appears extensive checks towards the involved solutions lead ambiguous results suggesting 'no clear mapping’ but typically confirm checking approachable with rational inputs leads understandably invalidated when found non-receptive against noted inputs.
Final responsiveness reflects upon the typical correct evaluation metrics leading standard math engagements.
3 answers