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A201905 Decimal expansion of the least x satisfying x^2+4x+2=e^x.

%I A201905 #9 Feb 07 2025 16:44:07
%S A201905 3,4,2,5,6,6,7,4,1,0,2,0,2,8,7,7,3,7,3,2,6,5,6,2,6,0,6,4,7,2,5,8,1,6,
%T A201905 6,9,7,8,2,7,3,5,7,2,6,1,7,3,3,2,3,3,5,5,5,3,6,6,6,3,4,3,8,0,6,5,1,2,
%U A201905 9,4,4,3,4,9,4,2,4,4,2,7,5,0,1,2,8,7,3,9,9,6,5,9,7,0,2,5,7,7,1
%N A201905 Decimal expansion of the least x satisfying x^2+4x+2=e^x.
%C A201905 See A201741 for a guide to related sequences. The Mathematica program includes a graph.
%H A201905 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201905 least:  -3.425667410202877373265626064725816697827357...
%e A201905 nearest to 0:  -0.35687491913863648565066705875991244...
%e A201905 greatest:  3.2349232177760663670327961327304430448478...
%t A201905 a = 1; b = 4; c = 2;
%t A201905 f[x_] := a*x^2 + b*x + c; g[x_] := E^x
%t A201905 Plot[{f[x], g[x]}, {x, -4, 3.3}, {AxesOrigin -> {0, 0}}]
%t A201905 r = x /. FindRoot[f[x] == g[x], {x, -3.5, -3.4}, WorkingPrecision -> 110]
%t A201905 RealDigits[r]     (* A201905 *)
%t A201905 r = x /. FindRoot[f[x] == g[x], {x, -.36, -.35}, WorkingPrecision -> 110]
%t A201905 RealDigits[r]     (* A201906 *)
%t A201905 r = x /. FindRoot[f[x] == g[x], {x, 3.2, 3.3}, WorkingPrecision -> 110]
%t A201905 RealDigits[r]     (* A201907 *)
%Y A201905 Cf. A201741.
%K A201905 nonn,cons
%O A201905 1,1
%A A201905 _Clark Kimberling_, Dec 06 2011