100000w^4-27000000w^2+10000000=0

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Solution for 100000w^4-27000000w^2+10000000=0 equation: 


Simplifying
100000w4 + -27000000w2 + 10000000 = 0

Reorder the terms:
10000000 + -27000000w2 + 100000w4 = 0

Solving
10000000 + -27000000w2 + 100000w4 = 0

Solving for variable 'w'.

Factor out the Greatest Common Factor (GCF), '100000'.
100000(100 + -270w2 + w4) = 0

Ignore the factor 100000.

Subproblem 1
Set the factor '(100 + -270w2 + w4)' equal to zero and attempt to solve:

Simplifying
100 + -270w2 + w4 = 0

Solving
100 + -270w2 + w4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-100' to each side of the equation.
100 + -270w2 + -100 + w4 = 0 + -100

Reorder the terms:
100 + -100 + -270w2 + w4 = 0 + -100

Combine like terms: 100 + -100 = 0
0 + -270w2 + w4 = 0 + -100
-270w2 + w4 = 0 + -100

Combine like terms: 0 + -100 = -100
-270w2 + w4 = -100

The w term is -270w2.  Take half its coefficient (-135).
Square it (18225) and add it to both sides.

Add '18225' to each side of the equation.
-270w2 + 18225 + w4 = -100 + 18225

Reorder the terms:
18225 + -270w2 + w4 = -100 + 18225

Combine like terms: -100 + 18225 = 18125
18225 + -270w2 + w4 = 18125

Factor a perfect square on the left side:
(w2 + -135)(w2 + -135) = 18125

Calculate the square root of the right side: 134.629120178

Break this problem into two subproblems by setting 
(w2 + -135) equal to 134.629120178 and -134.629120178.


Subproblem 1
w2 + -135 = 134.629120178

Simplifying
w2 + -135 = 134.629120178

Reorder the terms:
-135 + w2 = 134.629120178

Solving
-135 + w2 = 134.629120178

Solving for variable 'w'.

Move all terms containing w to the left, all other terms to the right.

Add '135' to each side of the equation.
-135 + 135 + w2 = 134.629120178 + 135

Combine like terms: -135 + 135 = 0
0 + w2 = 134.629120178 + 135
w2 = 134.629120178 + 135

Combine like terms: 134.629120178 + 135 = 269.629120178
w2 = 269.629120178

Simplifying
w2 = 269.629120178

Take the square root of each side:
w = {-16.420387333, 16.420387333}


Subproblem 2
w2 + -135 = -134.629120178

Simplifying
w2 + -135 = -134.629120178

Reorder the terms:
-135 + w2 = -134.629120178

Solving
-135 + w2 = -134.629120178

Solving for variable 'w'.

Move all terms containing w to the left, all other terms to the right.

Add '135' to each side of the equation.
-135 + 135 + w2 = -134.629120178 + 135

Combine like terms: -135 + 135 = 0
0 + w2 = -134.629120178 + 135
w2 = -134.629120178 + 135

Combine like terms: -134.629120178 + 135 = 0.370879822
w2 = 0.370879822

Simplifying
w2 = 0.370879822

Take the square root of each side:
w = {-0.608999033, 0.608999033}


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {-16.420387333, 16.420387333, -0.608999033, 0.608999033}
Solution
w = {-16.420387333, 16.420387333, -0.608999033, 0.608999033}

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