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22 Nov 2021, 06:26
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If \(3^{(2x+1)} + 4(3^x) – 20 = 0\), then what is approximate value of \(x\) ?
A. \(\frac{1}{6}\)
B. \(\frac{1}{5}\)
C. \(\frac{1}{3}\)
D. \(\frac{2}{3}\)
E. \(\frac{9}{10}\)
A. \(\frac{1}{6}\)
B. \(\frac{1}{5}\)
C. \(\frac{1}{3}\)
D. \(\frac{2}{3}\)
E. \(\frac{9}{10}\)
22 Nov 2021, 06:26
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Official Solution:
If \(3^{(2x+1)} + 4(3^x) – 20 = 0\), then what is approximate value of \(x\) ?
A. \(\frac{1}{6}\)
B. \(\frac{1}{5}\)
C. \(\frac{1}{3}\)
D. \(\frac{2}{3}\)
E. \(\frac{9}{10}\)
\(3^{(2x+1)} + 4(3^x) – 20 = 0\);
\(3*3^{(2x)} + 4(3^x) – 20 = 0\);
\(3*(3^x)^2 + 4(3^x) – 20 = 0\);
Denote \(3^x\) as \(k\): \(3k^2 + 4k– 20 = 0\);
Solve for \(k\): \(k=2\) or \(k=-\frac{10}{3}\) (discard this root because \(3^x\) is positive and cannot equal to a negative number).
We get \(3^x=2\)
Now, \(3^2 \approx 2^3\), so \(3^{\frac{2}{3}} \approx 2\), thus \(x\approx {\frac{2}{3}}\).
Answer: D
If \(3^{(2x+1)} + 4(3^x) – 20 = 0\), then what is approximate value of \(x\) ?
A. \(\frac{1}{6}\)
B. \(\frac{1}{5}\)
C. \(\frac{1}{3}\)
D. \(\frac{2}{3}\)
E. \(\frac{9}{10}\)
\(3^{(2x+1)} + 4(3^x) – 20 = 0\);
\(3*3^{(2x)} + 4(3^x) – 20 = 0\);
\(3*(3^x)^2 + 4(3^x) – 20 = 0\);
Denote \(3^x\) as \(k\): \(3k^2 + 4k– 20 = 0\);
Solve for \(k\): \(k=2\) or \(k=-\frac{10}{3}\) (discard this root because \(3^x\) is positive and cannot equal to a negative number).
We get \(3^x=2\)
Now, \(3^2 \approx 2^3\), so \(3^{\frac{2}{3}} \approx 2\), thus \(x\approx {\frac{2}{3}}\).
Answer: D
21 Jan 2025, 04:51
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I like the solution - it’s helpful.
