SOLUTION: Solve. {{{w^4-18w^2-2=0}}}

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Question 390276: Solve.
w%5E4-18w%5E2-2=0
Found 2 solutions by nerdybill, richard1234:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
w%5E4-18w%5E2-2=0
.
Let x = w%5E2
now, we can rewrite our equation as:
.
x%5E2-18x-2=0
applying the quadratic formula, we get:
x = {18.1104, -0.1104}
.
since w^2 = x
w = sqrt%28x%29
x = {4.256}
.
details of quadratic follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-18x%2B-2+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-18%29%5E2-4%2A1%2A-2=332.

Discriminant d=332 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--18%2B-sqrt%28+332+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-18%29%2Bsqrt%28+332+%29%29%2F2%5C1+=+18.1104335791443
x%5B2%5D+=+%28-%28-18%29-sqrt%28+332+%29%29%2F2%5C1+=+-0.110433579144299

Quadratic expression 1x%5E2%2B-18x%2B-2 can be factored:
1x%5E2%2B-18x%2B-2+=+1%28x-18.1104335791443%29%2A%28x--0.110433579144299%29
Again, the answer is: 18.1104335791443, -0.110433579144299. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-18%2Ax%2B-2+%29

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Letting z+=+w%5E2, substitute:

z%5E2+-+18z+-+2+=+0

By the quadratic formula, z+=+%2818+%2B-+sqrt%28332%29%29%2F2+=+9+%2B-+sqrt%2883%29 If z+=+9+%2B+sqrt%2883%29 then w+=+0+%2B-+sqrt%289+%2B+sqrt%2883%29%29. Likewise, if z+=+9+-+sqrt%2883%29 then w+=+0+%2B-+sqrt%289+-+sqrt%2883%29%29 (this gives two real and two complex values for w).