% C. Talarico
% file: improvedquadratic.m

clear all; close all; clc;
a = 1 ; b = 45000; c = 1;
Q_square = a*c/b^2
format shortE; % set display format

fprintf('--------------------------------\n');
fprintf('roots using standard format\n');
x1 = (- b + sqrt(b^2 - 4*a*c))/2
x2 = (- b - sqrt(b^2 - 4*a*c))/2
fprintf('--------------------------------\n');

fprintf('roots using improved format\n');
x1 = -b/2/a*(1 - sqrt(1-4*a*c/b^2))
x2 = -b/2/a*(1 + sqrt(1-4*a*c/b^2))
fprintf('--------------------------------\n');

fprintf('roots using approximation\n');
x2 = -b/a
x1 = (c/a)/x2
fprintf('--------------------------------\n');

fprintf('roots using symbolic matlab\n');
syms x;
y = a*x^2+b*x+c;
x12 = solve(y==0, x);
x1 = double(vpa(x12(1)))
x2 = double(vpa(x12(2)))
fprintf('--------------------------------\n');

% evaluating quality of approx
Q = 0:0.01:0.5;
F = 1 + (1 - 4*Q.^2).^0.5;
figure(1);
plot(Q,F,'linewidth',1)
ax = gca;
set(ax, 'Fontsize',14, 'FontName', 'Monospace', 'Fontweight', 'Normal');
xlabel('Q','fontsize',16,'fontname','Monospace','fontweight','Bold');
ylabel('F','fontsize',16,'fontname','Monospace','fontweight','Bold');

figure(2);
E = 100*(2 - F) ./ F;
plot(Q,E,'linewidth',2, 'color','r')
ax = gca;
set(ax, 'Fontsize',14, 'FontName', 'Monospace', 'Fontweight', 'Normal');
xlabel('Q','fontsize',16,'fontname','Monospace','fontweight','Bold');
title('% Error = 100 x (Approx Value - Exact Value)/Exact Value', ...
    'Fontsize',18);

Q1 = 0.3;
F1 = 1 + (1 - 4*Q1^2)^0.5;
E1 = 100*(2 - F1)/F1;
e_str = sprintf('   %% Error for Q \\leq 0.3 is \\leq %0.2f %%', E1);
leg = legend(e_str,'location','Best');
set(leg,'Box','off', 'fontsize',16);