6x^6 + 30x^5 + 36x^4 =0
divide through by 6 first
6(x^6+5x^5+6x^4)=0
take out the lowest common factor x^4
6x^4(x^2+5x+6)
then factorise the bracket
6x^4.(x+2)(x+3)
you can of course just do it in one step and take out 6x^4 if you are comfortable enough to do that
x = 0
x = -1
x = -6
??
or maybe not ... my calculator is broken
from Jordan's last line ...
6x^4(x+2)(x+3) = 0
x = 0
x = -2
x = -3
To solve the equation 6x^6 + 30x^5 + 36x^4 = 0, we can first notice that each term has a common factor of 6x^4. Factoring out 6x^4, we get:
6x^4(x^2 + 5x + 6) = 0
Now, we have two factors: 6x^4 and (x^2 + 5x + 6). In order for the whole equation to be equal to zero, at least one of the factors must be equal to zero.
Setting 6x^4 = 0, we find that x = 0.
Setting x^2 + 5x + 6 = 0, we can solve this quadratic equation by factoring or using the quadratic formula:
(x + 2)(x + 3) = 0
So, x = -2 or x = -3.
Therefore, the equation 6x^6 + 30x^5 + 36x^4 = 0 has three solutions: x = 0, x = -2, and x = -3.