SOLUTION: Real or imaginary solutions x^3=216 6x=9+x^2 X^2+3x+3=0 X^3+125=0 X^4-24x^2+100=0

Question 840889: Real or imaginary solutions
x^3=216
6x=9+x^2
X^2+3x+3=0
X^3+125=0
X^4-24x^2+100=0
Found 2 solutions by Fombitz, KMST:
Answer by Fombitz(32388) (Show Source): You can put this solution on YOUR website!
Both
Real
Imaginary
Both
Real
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
-->-->-->-->
, which makes is the real solution;
, which has no real solution yields two more imaginary ones.

-->--> has as a double real solution (no imaginary ones).

has no real solutions, just imaginary ones.
You would know that by calculating the discriminant.
A quadratic equation, that can be written as
, has solutions that can be calculated using the quadratic formula:

The expression is called the discriminant, and its sign will tell you about the solutions. A negative discriminant means no real solutions. A positive one means two (different) real solutions. If the discriminant is zero, there is just one (double) real solution (as happened for above).
In this case , , and , and
, so no real solutions.

-->-->
is the real solution; , which has no real solution yields two more imaginary ones.

is more complicated.
You could use the change of variable to write the equation as
, and find that the solutions for .
You could solve for by completing the square, or by using the quadratic formula.
Either way, you will find
(or an equivalent expression).
Since both solutions are positive, you will have four real solutions to
, and those are the solutions of
.
so there are two real solutions.

RELATED QUESTIONS
find all real and imaginary roots... (answered by stanbon,Edwin McCravy)
real or imaginary solutions to... (answered by jim_thompson5910)
What are real or imaginary solutions of the polynomials.... (answered by Boreal,josmiceli)
give all solutions to... (answered by htmentor)
Factorize y = 0 0 = X^3+3x^2-24x-16 (answered by Fombitz)
Find all real or imaginary solutions to the equation.... (answered by jim_thompson5910)
find all real or imaginary solutions to each equation. x-1/x+2 = 2x-3/x+4 (answered by jim_thompson5910)
find all real solutions of the equation.... (answered by jim_thompson5910)
Solve using substitution. Find all solutions, real or complex. {{{ x^(4/3) - 3x^(2/3)... (answered by Alan3354)