SOLUTION: Solve. w^4-20w^2-2=0 What is the solution?

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Solve. w^4-20w^2-2=0 What is the solution?      Log On


   

Question 632148: Solve.
w^4-20w^2-2=0
What is the solution?
Found 2 solutions by josmiceli, Edwin McCravy:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +w%5E4+=+z%5E2+
+z+=+w%5E2+ and
+z+=+-w%5E2+
+w%5E4+-+20w%5E2+-+2+=+0++
+z%5E2+-+20z++-+2+=+0+
+z%5E2+-+20z+=+2+
Complete the square
+z%5E2+-+20z+%2B+%28-20%2F2%29%5E2+=+2+%2B+%28-20%2F2%29%5E2+
+z%5E2+-+20z+%2B+100++=+2+%2B+100+
+%28+z+-+10+%29%5E2+=+102+
Take the square root of both sides
+z+-+10+=+sqrt%28102%29+
+z+=+10+%2B+sqrt%28+102+%29+
and
+z+=+10+-+sqrt%28+102+%29+
----------------------
+w%5E2+=+10+%2B+sqrt%28+102+%29+
+w+=+sqrt%28+10+%2B+sqrt%28102%29+%29+
+w+=+-sqrt%28+10+%2B+sqrt%28102%29+%29+
-----------------------
+w%5E2+=+10+-+sqrt%28102%29+
+w+=+sqrt%28+10+-+sqrt%28102%29+%29+
+w+=+-sqrt%28+10+-+sqrt%28102%29+%29+
-----------------------
These are the 4 roots
Here's a plot of the equation:
+graph%28+400%2C+400%2C+-8%2C+8%2C+-120%2C+20%2C+x%5E4+-+20x%5E2+-+2+%29+
There are 2 real roots and 2 imaginary, which
seems to agree with the plot

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
I just might add that the two imaginary solutions
+w+=+%22%22+%2B-+sqrt%28+10+-+sqrt%28102%29+%29+
can be written as
+w+=+%22%22+%2B-+sqrt%28-1%2A%28-10+%2B+sqrt%28102%29%29+%29+
+w+=+%22%22+%2B-+i%2Asqrt%28-10+%2B+sqrt%28102%29+%29+
+w+=+%22%22+%2B-+i%2Asqrt%28sqrt%28102%29-10+%29+
Edwin