Algebra

Zozo:

Determine the values of x for the following x^4 - 6x^2 +6 = x^2+ 6x +6 .?

i'd start by taking it all to one side.

x ⁴ - 7x ² - 6x = 0
Then i'd do any easy factorising

x(x ³ - 7x - 6) = 0
now I can see that one answer is x=0

to get the other answers, and there could be 3 of them (because of the x ³)
i would try to factorise 1x ³ - 7x - 6

Any integer root will be a factor of -6/1 = -6 that is, it could be 1, -1, 2, -2, 3, -3, 6, -6 there are no other choices. (The root could of course be a non integer)

f(x) = x ³ - 7x - 6
f(1) = 1-7-6 not equal to 0 doesn't help
f(-1) = -1+7-6 = 0 therefore (x+1) is a factor and x=-1 is a solution to the original equation.
at this point I could do an algebraic division before i look for the next root but i might just continue here and see if I can find another one easily

f(2)=8-14-6 not equal to zero
f(-2)= -8+14-6 = 0 therefore (x+2) is a factor and x=-2 is a solution to the original equation
Again i could do an anlgebraic division but I'm going to see if I can continue my lucky streak here.

f(3)=27-21-6=0 therefore (x-3) is a factor and x=3 is a solution to the original equation.
that's it, i knew that there could not be more than 3 more roots. so I have them all.

So the 4 roots are x=0, x=-1, x=-2, and x=3

If I graphed y = x ⁴ - 7x ² - 6x this is what I get.