x^4-25x^2+135=0

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Solution for x^4-25x^2+135=0 equation:


Simplifying
x4 + -25x2 + 135 = 0

Reorder the terms:
135 + -25x2 + x4 = 0

Solving
135 + -25x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-135' to each side of the equation.
135 + -25x2 + -135 + x4 = 0 + -135

Reorder the terms:
135 + -135 + -25x2 + x4 = 0 + -135

Combine like terms: 135 + -135 = 0
0 + -25x2 + x4 = 0 + -135
-25x2 + x4 = 0 + -135

Combine like terms: 0 + -135 = -135
-25x2 + x4 = -135

The x term is -25x2.  Take half its coefficient (-12.5).
Square it (156.25) and add it to both sides.

Add '156.25' to each side of the equation.
-25x2 + 156.25 + x4 = -135 + 156.25

Reorder the terms:
156.25 + -25x2 + x4 = -135 + 156.25

Combine like terms: -135 + 156.25 = 21.25
156.25 + -25x2 + x4 = 21.25

Factor a perfect square on the left side:
(x2 + -12.5)(x2 + -12.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(x2 + -12.5) equal to 4.609772229 and -4.609772229.

Subproblem 1

x2 + -12.5 = 4.609772229 Simplifying x2 + -12.5 = 4.609772229 Reorder the terms: -12.5 + x2 = 4.609772229 Solving -12.5 + x2 = 4.609772229 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12.5' to each side of the equation. -12.5 + 12.5 + x2 = 4.609772229 + 12.5 Combine like terms: -12.5 + 12.5 = 0.0 0.0 + x2 = 4.609772229 + 12.5 x2 = 4.609772229 + 12.5 Combine like terms: 4.609772229 + 12.5 = 17.109772229 x2 = 17.109772229 Simplifying x2 = 17.109772229 Take the square root of each side: x = {-4.136396044, 4.136396044}

Subproblem 2

x2 + -12.5 = -4.609772229 Simplifying x2 + -12.5 = -4.609772229 Reorder the terms: -12.5 + x2 = -4.609772229 Solving -12.5 + x2 = -4.609772229 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12.5' to each side of the equation. -12.5 + 12.5 + x2 = -4.609772229 + 12.5 Combine like terms: -12.5 + 12.5 = 0.0 0.0 + x2 = -4.609772229 + 12.5 x2 = -4.609772229 + 12.5 Combine like terms: -4.609772229 + 12.5 = 7.890227771 x2 = 7.890227771 Simplifying x2 = 7.890227771 Take the square root of each side: x = {-2.808954925, 2.808954925}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.136396044, 4.136396044, -2.808954925, 2.808954925}

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