Question #7f145

This question can be solved via treating the function # x^4 - x^2 +6 # as a quadratic...

The first thing we can do is make a simple substitution:

#psi = x^2 #

Where #psi # is just a random variable i have choosen, you choose what ever you like

#=> psi^2 = x^4 #

Hence our original function transforms into:

#psi^2 - psi + 6 = 0 #

Now we can just solve like we ussually do:

We can just use the quadratic equation:

#psi = (-b pm sqrt( b^2-4ac) ) /(2a) #

For # apsi^2 + bpsi + c = 0 #

#psi = (-(-1) pm sqrt( (-1)^2 - (4*1*6) ) )/(2*1) #

#=> psi = (1 pm sqrt(-23) )/2 #

We know form our complex number studies that #i = sqrt(-1) #

#=> psi = x^2 = ( 1 pm isqrt(23) ) /2 #

# => x =pm sqrt (( 1 pm isqrt(23) ) /2 ) #

#color(red)(=> x= { pm sqrt(( 1 + isqrt(23) )/2) , pm sqrt(( 1 - isqrt(23) )/2)} #

We see by a sketch that there are no solutions for # x in RR # , or in other words, there are only solutions in the complex relm

If # y = ax^(2n) + bx^n + c #

Then we can make a substitution:

#phi = x^n #

#=> phi^2 = (x^n)^2 = x^(2n) #

Hence our equation can be transformed into:

# y = aphi^2 + bphi + c #

What we know is much more simpler to deal with