How do you find all the real and complex roots of #2x^4 + 3x^3 - x^2 + 5 = 0#?
1 Answer
See explanation for a sketch of how to solve this algebraically...
Explanation:
This is an example of the worst possible case in quartic equations.
The full solution is too long, so I will just sketch it here...
- Use a Tschirnhaus transformation to simplify the quartic to one with no cube term:
#2048f(x) = 4096x^4+6144x^3-2048x^2+10240#
#color(white)(2048f(x)) = (8x+3)^4-86(8x+3)^2+408(8x+3)+9709#
#color(white)(2048f(x)) = t^4-86t^2+408t+9709# where
#t = 8x+3#
- Consider a factorisation of the form
#(t^2-at+b)(t^2+at+c)# and equate coefficients to get:
#{ (b+c = a^2-86), (b-c = 408/a), (bc = 9709) :}#
- Use
#(b+c)^2 = (b-c)^2+4bc# to derive a cubic in#a^2# :
#(a^2)^3-172(a^2)^2-31440(a^2)-166464 = 0#
- Use a Tschirnhaus transformation to simplify the cubic to one with no square term:
#27((a^2)^3-172(a^2)^2-31440(a^2)-166464)#
#= 27(a^2)^3-46644(a^2)^2-848880(a^2)-4494528#
#= (3a^2-172)^3-371712(3a^2-172)-63340544#
#= s^3-371712s-63340544# where
#s = 3a^2-172#
- This cubic has
#3# Real zeros, so use a trigonometric substitution of the form#s = k cos theta# with#k=704#
#0 = s^3-371712s-63340544#
#color(white)(0) = 704^3 cos^3 theta - 371712*704 cos theta - 63340544#
#color(white)(0) = 87228416(4 cos^3 theta - 3cos theta) - 63340544#
#color(white)(0) = 87228416 cos 3 theta - 63340544#
#color(white)(0) = 32768(2662 cos 3 theta - 1933)# Hence:
#s = 704 cos(1/3cos^(-1)(1933/2662)+(2kpi)/3)# for#k=0,1,2# The positive Real root occurs for
#k=0# , so choose that to get:
#a = sqrt(1/3( 704 cos(1/3cos^(-1)(1933/2662))+172)#
- Going back to our simultaneous equations in
#a, b# and#c# we find:
#b = 1/2(a^2-86+408/a)#
#c = 1/2(a^2-86-408/a)#
- Hence we have two quadratic equations to solve:
#t^2-at+1/2(a^2-86+408/a) = 0#
#t^2+at+1/2(a^2-86-408/a) = 0#
-
Solve these quadratic equations using the quadratic formula to find solutions to
#t^4-86t^2+408t+9709=0# -
Reverse the initial Tschirnhaus transformation using
#x = 1/8(t-3)# to find the four (Complex) roots of the original quartic.
