x^4-15x^2+35=0

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Solution for x^4-15x^2+35=0 equation:


Simplifying
x4 + -15x2 + 35 = 0

Reorder the terms:
35 + -15x2 + x4 = 0

Solving
35 + -15x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-35' to each side of the equation.
35 + -15x2 + -35 + x4 = 0 + -35

Reorder the terms:
35 + -35 + -15x2 + x4 = 0 + -35

Combine like terms: 35 + -35 = 0
0 + -15x2 + x4 = 0 + -35
-15x2 + x4 = 0 + -35

Combine like terms: 0 + -35 = -35
-15x2 + x4 = -35

The x term is -15x2.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15x2 + 56.25 + x4 = -35 + 56.25

Reorder the terms:
56.25 + -15x2 + x4 = -35 + 56.25

Combine like terms: -35 + 56.25 = 21.25
56.25 + -15x2 + x4 = 21.25

Factor a perfect square on the left side:
(x2 + -7.5)(x2 + -7.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(x2 + -7.5) equal to 4.609772229 and -4.609772229.

Subproblem 1

x2 + -7.5 = 4.609772229 Simplifying x2 + -7.5 = 4.609772229 Reorder the terms: -7.5 + x2 = 4.609772229 Solving -7.5 + x2 = 4.609772229 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x2 = 4.609772229 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x2 = 4.609772229 + 7.5 x2 = 4.609772229 + 7.5 Combine like terms: 4.609772229 + 7.5 = 12.109772229 x2 = 12.109772229 Simplifying x2 = 12.109772229 Take the square root of each side: x = {-3.479909802, 3.479909802}

Subproblem 2

x2 + -7.5 = -4.609772229 Simplifying x2 + -7.5 = -4.609772229 Reorder the terms: -7.5 + x2 = -4.609772229 Solving -7.5 + x2 = -4.609772229 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x2 = -4.609772229 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x2 = -4.609772229 + 7.5 x2 = -4.609772229 + 7.5 Combine like terms: -4.609772229 + 7.5 = 2.890227771 x2 = 2.890227771 Simplifying x2 = 2.890227771 Take the square root of each side: x = {-1.70006699, 1.70006699}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.479909802, 3.479909802, -1.70006699, 1.70006699}

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