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):
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.
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