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A201934 Decimal expansion of the x nearest 0 that satisfies x^2+5x+2=e^x.

%I A201934 #10 Mar 30 2025 16:01:22
%S A201934 2,5,9,0,6,9,5,3,3,0,5,1,1,0,9,1,0,8,6,8,6,4,0,5,6,6,4,6,5,5,9,6,2,2,
%T A201934 6,2,8,9,6,4,8,0,5,4,5,7,8,6,4,2,5,5,1,3,1,6,9,2,1,5,6,5,9,4,9,0,1,7,
%U A201934 2,4,9,0,0,0,8,8,2,5,6,7,1,2,6,4,9,8,1,3,4,8,3,9,7,0,1,2,4,8,4
%N A201934 Decimal expansion of the x nearest 0 that satisfies x^2+5x+2=e^x.
%C A201934 See A201741 for a guide to related sequences. The Mathematica program includes a graph.
%H A201934 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201934 least:  -4.5640783603793772013414868523420...
%e A201934 nearest to 0:  -0.259069533051109108686405...
%e A201934 greatest:  3.43200871161068035280379146269...
%t A201934 a = 1; b = 5; c = 2;
%t A201934 f[x_] := a*x^2 + b*x + c; g[x_] := E^x
%t A201934 Plot[{f[x], g[x]}, {x, -5, 3.5}, {AxesOrigin -> {0, 0}}]
%t A201934  r = x /. FindRoot[f[x] == g[x], {x, -4.6, -4.5}, WorkingPrecision -> 110]
%t A201934 RealDigits[r]     (* A201933 *)
%t A201934  r = x /. FindRoot[f[x] == g[x], {x, -.3, -.2}, WorkingPrecision -> 110]
%t A201934 RealDigits[r]     (* A201934 *)
%t A201934  r = x /. FindRoot[f[x] == g[x], {x, 3.4, 3.5}, WorkingPrecision -> 110]
%t A201934 RealDigits[r]     (* A201935 *)
%t A201934 RealDigits[x/.FindRoot[x^2+5x+2==E^x,{x,1},WorkingPrecision->120],10,120][[1]] (* _Harvey P. Dale_, Mar 30 2025 *)
%Y A201934 Cf. A201741.
%K A201934 nonn,cons
%O A201934 0,1
%A A201934 _Clark Kimberling_, Dec 06 2011