Solve -x^3+2x^2+9x+22

Question

I need to find the real and complex solutions.

Answer 1

Solve –x^3 + 2x^2 + 9x + 22
I need to find the real and complex solutions.

This is a cubic equation, with real and complex solutions. There will be two complex solutions and one real solution.

The real solution
Use the Newton-Raphson method on f(x)= -x^3+2x^2+9x+22
The N-R formula to use is:

x_(n+1)=x_n - f(x_n )/(f^' (x_n ) )

Where f(x)= -x^3+2x^2+9x+22,  f^' (x)=-3x^2+4x+9, and x_1=4. (found by trial and error)

n |     x_n    |       f(x_n)        |  f^' (x_n ) | x_(n+1)=x_n-f(x_n )/(f^' (x_n ) )
1 |   4          |      26              |   -23        | 5.13043
2 | 5.13043 | -14.2234         | -49.4423 | 4.84276
3 | 4.84276 | -1.08443         | -41.9859 | 4.81693
4 | 4.81693 | -8.3404 E -03 | -41.3407 | 4.81673
5 | 4.81673 | -5.0677 E -07 | -41.3357 | 4.81673

So, to 5 d.p. our real solution is: x=4.81673

Our original expression is: -x^3+2x^2+9x+22. Taking out a factor of (x-4.81673), we get

(x - 4.81673)(-x^2 - 2.81673x - 4.567416)

The remaining quadratic is: g(x)=-x^2 - 2.81673x - 4.567416.
Solving this with the quadratic formula,
x=(-b±√(b^2-4ac))/2a
With a=-1, b=-2.81673, c=-4.567416. Then,

x=(2.81673±√(2.81673^2 - 4(-1)(-4.567416) ))/(2*(-1) )
x=(2.81673±√(7.93396 - 18.26966))/(-2)
x=(2.81673±i√10.33571)/(-2)
x = -1.40836 ± 1.60746i