15x^4-15x^2+1=0

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Solution for 15x^4-15x^2+1=0 equation:


Simplifying
15x4 + -15x2 + 1 = 0

Reorder the terms:
1 + -15x2 + 15x4 = 0

Solving
1 + -15x2 + 15x4 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
15 the coefficient of the squared term: 

Divide each side by '15'.
0.06666666667 + -1x2 + x4 = 0

Move the constant term to the right:

Add '-0.06666666667' to each side of the equation.
0.06666666667 + -1x2 + -0.06666666667 + x4 = 0 + -0.06666666667

Reorder the terms:
0.06666666667 + -0.06666666667 + -1x2 + x4 = 0 + -0.06666666667

Combine like terms: 0.06666666667 + -0.06666666667 = 0.00000000000
0.00000000000 + -1x2 + x4 = 0 + -0.06666666667
-1x2 + x4 = 0 + -0.06666666667

Combine like terms: 0 + -0.06666666667 = -0.06666666667
-1x2 + x4 = -0.06666666667

The x term is -1x2.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x2 + 0.25 + x4 = -0.06666666667 + 0.25

Reorder the terms:
0.25 + -1x2 + x4 = -0.06666666667 + 0.25

Combine like terms: -0.06666666667 + 0.25 = 0.18333333333
0.25 + -1x2 + x4 = 0.18333333333

Factor a perfect square on the left side:
(x2 + -0.5)(x2 + -0.5) = 0.18333333333

Calculate the square root of the right side: 0.428174419

Break this problem into two subproblems by setting 
(x2 + -0.5) equal to 0.428174419 and -0.428174419.

Subproblem 1

x2 + -0.5 = 0.428174419 Simplifying x2 + -0.5 = 0.428174419 Reorder the terms: -0.5 + x2 = 0.428174419 Solving -0.5 + x2 = 0.428174419 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x2 = 0.428174419 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = 0.428174419 + 0.5 x2 = 0.428174419 + 0.5 Combine like terms: 0.428174419 + 0.5 = 0.928174419 x2 = 0.928174419 Simplifying x2 = 0.928174419 Take the square root of each side: x = {-0.963418091, 0.963418091}

Subproblem 2

x2 + -0.5 = -0.428174419 Simplifying x2 + -0.5 = -0.428174419 Reorder the terms: -0.5 + x2 = -0.428174419 Solving -0.5 + x2 = -0.428174419 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x2 = -0.428174419 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = -0.428174419 + 0.5 x2 = -0.428174419 + 0.5 Combine like terms: -0.428174419 + 0.5 = 0.071825581 x2 = 0.071825581 Simplifying x2 = 0.071825581 Take the square root of each side: x = {-0.26800295, 0.26800295}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.963418091, 0.963418091, -0.26800295, 0.26800295}

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