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\( 2x^2 + 6x + 9 = x^2 + 10x + 25 \)
\( x^2 - 4x - 16 = 0 \)
Use quadratic formula: \( x = \frac{4 \pm \sqrt{16 + 64}}{2} = \frac{4 \pm \sqrt{80}}{2} \)
\( x = \frac{4 \pm 4\sqrt{5}}{2} = 2 \pm 2\sqrt{5} \)
Only positive value: \( x = 2 + 2\sqrt{5} \)
#### \( 2 + 2\sqrt{5} \)
The population of a city grows exponentially according to \( P(t) = 50000 \cdot e^{0.03t} \), where \( t \) is in years. How long will it take for the population to double?
Set \( P(t) = 100000 = 50000 \cdot e^{0.03t} \)
\( 2 = e^{0.03t} \)
Take natural log: \( \ln 2 = 0.03t \)