a nice exponential equation olympiad math | how to solve for x |

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[@MUNEEBURREHMANMianMuneeb]

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[@ABCricinfofan-el4ju]

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[@SaimaBaloch-e9t]

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[@m.safdarm.safdar277]

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[@zaidsarfraz-b3y]

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[@B0rschec]

x = 1
did this with little calculations

[@oahuhawaii2141]

Let y = 2ˣ . For x ∈ ℝ , y ∈ ℝ and y > 0 .
y³ + y² = 12
y*y*(y + 1) = 2*2*(2 + 1)
y = 2 -- factor out (y - 2)
y³ + y² - 12 = (y - 2)*(y² + 3*y + 6) = 0
y = 2, (-3 ± i*√15)/2 { Discard y ∉ ℝ , y ≤ 0 }
2ˣ = 2
x = 1
2ˣ = (-3 ± i*√15)/2 = r*e^(i*θ)
r = √6 , θ = ±arccos(-√6/4) + 2*π*k , k ∈ ℤ
x = ln(6)/ln(2)/2 + i*(2*π*k ± arccos(-√6/4))/ln(2)