Question

The following procedure can be used to determine the roots of a cubic equation a_3x^3 + a_2x^2 + a_1x + a_0 = 0: Set: A =a_2/a_3, B = a_1/a_3, and C = a_0/a_3 Calculate: D = Q^3 + R^2 where Q = (3B - A^2)/9 and R = (9AB - 27C - 2A^3)/54. If D > 0, the equation has complex roots. It D = 0, all roots are real and at least two are equal. The roots are given by: x_1=2 cubicroot R - A/3, and x_2 = x_3 = - cubicroot R - A/3. If D < 0, all roots are real and are given by: x_1 = 2 Squareroot -Q cos(theta/3) - A/3, x_2 = 2Squareroot -Q cos(theta/3 + 120 degree) -A/3, and x_3 = 2Squareroot -Q cos(theta/3 + 240 degree) - A/3, where cos theta = R/Squareroot -Q^3 Write a MATLAB program in a script file that determines the real roots of a cubic equation. As input, the program asks the user to enter the values of a_3, a_2, a_1 and a_0 as a vector. The program then calculates the value of D. If the equation has complex roots the message "The equation CCCx^3 + CCCx^2 + CCCx + CCC has complex roots" is displayed. Otherwise the real roots are calculated and displayed as "The roots of the equation CCCx^3 + CCCx^2 + CCCx + CCC are XXX, XXX, and XXX". Where CCC refer to the value of the corresponding coefficient, + refer to the sign of the coefficient (+ or -) and XXX refer to the value of the roots. Use fprintf () to display the text and data. The data should be displayed in g format. Use the program to determine the roots of the following equations: 5x^3 - 34.5 x^2 + 36.9 x + 8.8 = 0 2x^3 - 10 x^2 + 24 x - 15 = 0 x^3 - 9 x^2 + 24 x - 20 = 0

Answer 1

Matlab code to find the roots of cubic equation and the code has to be written in script file:-

The code is explained in the form of comments i.e followed by the symbol %

prompt='enter the values of a3,a2,a1,a0 respectively \n'; % to display the message to user
x=input(prompt); % input entered by the user will be stored in x in the form of vector
a3=x(1); % initializing a3
a2=x(2); % initializing a2
a1=x(3); % initializing a1
a0=x(4); % initializing a0
A=a2/a3; % assigning the value of A as given by the formula for A in the given problem
B=a1/a3; % assigning the value of B as given by the formula for B in the given problem
C=a0/a3; % assigning the value of C as given by the formula for C in the given problem
Q=(3*B-A*A)/9; % assigning the value of Q as given by the formula for Q in the given problem
R=(9*A*B-27*C-2*A*A*A)/54; % assigning the value of R as given by the formula for R in the given problem
D=(Q*Q*Q)+(R*R); % assigning the value of D as given by the formula for D in the given problem
formatspec='The equation %4.2fx^3+%4.2fx^2+%4.2fx+%4.2f has complex roots \n';
% formatspecifier to set the format for complex roots
formatspec1='The roots of the equation %4.2fx^3 + %4.2fx^2 + %4.2fx + %4.2f are %4.2f , %4.2f and %4.2f \n';
% formatspecifier to set the format for other than complex roots
if D>0 % for D>0
fprintf(formatspec,x); % print that there are complex roots
end
if D==0 % for D==0
x1=(2*R^(1/3))-(A/3); % caluculate x1
x2= -(R^(1/3))-(A/3); % caluculate x2
x3=x2; % x2 and x3 are equal
fprintf(formatspec1,a3,a2,a1,a0,x1,x2,x3); % print the roots
end
if D<0
t=acosd(R/((-Q)^(3/2))); % caluculate theta
x1=2*((-Q)^(1/2))*(cosd(t/3))-(A/3); % caluculate x1
x2=2*((-Q)^(1/2))*(cosd(t/3+120))-(A/3); % caluculate x2
x3=2*((-Q)^(1/2))*(cosd(t/3+240))-(A/3); % caluculate x3
fprintf(formatspec1,a3,a2,a1,a0,x1,x2,x3); % print the roots
end
  
Output:-

After writing the above code run the program and enter the values for a3,a2,a1 and a0 in the command prompt for the given respective 3 equations to get the following output.

Command Window >> roots enter the values of a3,a2,al, ao respectively [5 -34.5 36.9 8.8 The roots of the equation 5.00x*3-34.50x 2+36.90x +8.80 are 5.50,-0.20 and 1.60 >>roots enter the values of a3,a2,a1,a0 respectively 2 -10 24 -15] The equation 2.00x^3+-10.00x^2+24.00x+-15.00 has complex roots >>roots enter the values of a3, a2, al,ao respectively [1 -9 24 -20] The roots of the equation 1.00x3 -9.00x*2 24.00x -20.00 are 5.00, 2.00 and 2.00 s>