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A201899 Decimal expansion of the greatest x satisfying x^2+3x+2=e^x.

%I A201899 #16 Jan 30 2025 11:34:07
%S A201899 2,9,9,2,2,3,4,8,7,2,0,5,3,9,3,6,8,6,5,0,9,3,3,1,1,4,5,2,7,8,3,8,8,2,
%T A201899 6,2,1,8,1,1,5,9,4,5,4,7,7,4,9,0,0,6,3,6,3,9,1,2,5,6,2,3,9,9,9,3,6,1,
%U A201899 6,8,9,8,5,4,9,6,4,7,1,9,5,1,2,1,1,4,9,4,4,6,8,2,5,6,7,1,0,5,1
%N A201899 Decimal expansion of the greatest x satisfying x^2+3x+2=e^x.
%C A201899 See A201741 for a guide to related sequences. The Mathematica program includes a graph.
%H A201899 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201899 least:  -2.1093569955710161272316992470592578841155...
%e A201899 nearest to 0:  -0.608989103010165494835043701926011...
%e A201899 greatest:  2.99223487205393686509331145278388262181...
%t A201899 a = 1; b = 3; c = 2;
%t A201899 f[x_] := a*x^2 + b*x + c; g[x_] := E^x
%t A201899 Plot[{f[x], g[x]}, {x, -3, 3.1}, {AxesOrigin -> {0, 0}}]
%t A201899 r = x /. FindRoot[f[x] == g[x], {x, -2.2, -2.1}, WorkingPrecision -> 110]
%t A201899 RealDigits[r]     (* A201897, least *)
%t A201899 r = x /. FindRoot[f[x] == g[x], {x, -.7, -.6}, WorkingPrecision -> 110]
%t A201899 RealDigits[r]     (* A201898, nearest 0  *)
%t A201899 r = x /. FindRoot[f[x] == g[x], {x, 2.9, 3.0}, WorkingPrecision -> 110]
%t A201899 RealDigits[r]     (* A201899 greatest *)
%Y A201899 Cf. A201741, A201897, A201898.
%K A201899 nonn,cons
%O A201899 1,1
%A A201899 _Clark Kimberling_, Dec 06 2011
%E A201899 Name corrected by _Sean A. Irvine_, Jan 12 2025