Can anyone show me an ALGEBRAIC solution?

Question

2n³-n²+48n-145=0

This equation is the resolvent cubic from a quartic equation I am solving.

It is easy to find the only real root graphically using DESMOS.

Can anyone show me an algebraic solution?

P.S. I am not looking for an approximation method (e.g. Newton's method); I am looking for a solution is so-called closed form.

Answer 1

You can solve the equation algebraically using factor by grouping. You need to manipulate the equation so that you can pull out a linear equation to solve.

2n³-n²+48n-145=0

2n³+4n²+58n-5n²-10n-145=0 Manipulate the coefficients for the n² and n, net difference stays the same

(2n³+4n²+58n)+(-5n²-10n-145)=0 Group the terms so you can factor out values

2n(n²+2n+29)-5(n²+2n+29)=0 Factor out values to simplify the terms

(2n-5)(n²+2n+29)=0 Now separate your equation that you found 2n-5 and set it equal to 0

2n-5=0

2n=5

n=5/2 or 2.5 This is your real zero

You can use the quadratic formula to find the other 2 zeros in the equation n²+2n+29

x= -b+sqrt((b²-4ac)/2a) and x= -b-sqrt((b²-4ac)/2a)

Answer 2

You could either use the cubic formula or the fact that all rational solutions must be of the form p/q, where p is a +/- factor of the constant term (-145 in this case) and q is a +/- factor of the leading coefficient (2 in this case). Synthetic division (or substitution) could then be used to verify the real solution of 5/2.

Here is a link to the cubic formula:

math.vanderbilt.edu/schectex/courses/cubic/

The synthetic division approach seems simpler to me.