A nice Lambert W Function Problem | Math Olympiad Algebra | Power Simplification Question | 27^x+x=0

[@UKPEINDANIELU.]

Fantastic

[@superacademy247]

Thank you so much 😀

[@gregevgeni1864]

Let f(x) = 27^x + x .
Easy to prove that f is strictly increasing function. (1)
But x=-1/3 is a obvious solution of the equation f(x)=0 (2).
From (1) and (2) the equation f(x)=0 have the unique solution, x=-1/3 .

[@nikolayguzman331]

More Faster: 27^X=-X If 27^X> 0 Than 0

[@walterwen2975]

Math Olympiad Algebra: 27ˣ + x = 0; x = ?
27ˣ > 0 > x; 27ˣ = - x
Trial-and-error math solution:
x = 0: 27⁰ = 1 > 0
x = - 1: 1/27 < - (- 1) = 1, 0 > x > - 1; In-between
x = - 1/2: 1/√27 = 1/(3√3) < - (- 1/2) = 1/2; 0 > x > - 1/2, Slightly > - 1/2
x = - 1/3: 1/(³√27) = 1/3 = - (- 1/3); Proved
Answer check:
27ˣ + x = 27⁻¹⸍³ + (- 1/3) = 1/3 - 1/3 = 0; Confirmed
Final answer:
x = - 1/3

[@superacademy247]

You should use a formula method instead of trial and error