Chapter 6, Problem 36P

The roots of a cubic equation $a_3{x}^3+a_2{x}^2+a_{1}x+a_0=0$ can be calculated using the following procedure: Set: A=a₂/a₃, B=a₁/ a₃, and C= a₀/ a₃.

Calculate: D =O³+R²,

where Q =(3B-A²)/9 and R(9AB027C-2A³)/54.

If D>0 the equation has complex roots.

If D=0 all roots are real and at least two are equal. The roots are given by:

$x_1=2\sqrt[3]{R}-A/3$ , $x_2=-\sqrt[3]{R}-A/3$and $x_3=-\sqrt[3]{R}-A/3$ .

If D<0 all roots are real and are given by:

$x_1=2\sqrt{-Q\cos (\theta }/3)-A/3$,

$x_2=2\sqrt{-Q\cos (\theta /}3+120°)-A/3$, and $x_3=2\sqrt{-Q\cos (\theta }/3+240°)-A/3$,where

$\theta =R/\sqrt{-Q^3}$.

Write a MATLAB program that determines the real roots of a cubic equation. As input the program asks the user to enter the values of a₃, a₂, a₁, and a₀as a vector. The program then calculates the value of D. If the equations have complex roots, the message “The equation has complex roots” is displayed. Otherwise the real roots are calculated and displayed. Use the program to solve the following equations:

(a)

$5x^3+-34.5x^2+36.9x+8.8=0$(b)

$2x^3+-10x^2+24x-15=0$(c)

$2x^3+-1.4x^2-20.58x+30.87=0$