5x^4-15x^2+4=0

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Solution for 5x^4-15x^2+4=0 equation:


Simplifying
5x4 + -15x2 + 4 = 0

Reorder the terms:
4 + -15x2 + 5x4 = 0

Solving
4 + -15x2 + 5x4 = 0

Solving for variable 'x'.

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

Divide each side by '5'.
0.8 + -3x2 + x4 = 0

Move the constant term to the right:

Add '-0.8' to each side of the equation.
0.8 + -3x2 + -0.8 + x4 = 0 + -0.8

Reorder the terms:
0.8 + -0.8 + -3x2 + x4 = 0 + -0.8

Combine like terms: 0.8 + -0.8 = 0.0
0.0 + -3x2 + x4 = 0 + -0.8
-3x2 + x4 = 0 + -0.8

Combine like terms: 0 + -0.8 = -0.8
-3x2 + x4 = -0.8

The x term is -3x2.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x2 + 2.25 + x4 = -0.8 + 2.25

Reorder the terms:
2.25 + -3x2 + x4 = -0.8 + 2.25

Combine like terms: -0.8 + 2.25 = 1.45
2.25 + -3x2 + x4 = 1.45

Factor a perfect square on the left side:
(x2 + -1.5)(x2 + -1.5) = 1.45

Calculate the square root of the right side: 1.204159458

Break this problem into two subproblems by setting 
(x2 + -1.5) equal to 1.204159458 and -1.204159458.

Subproblem 1

x2 + -1.5 = 1.204159458 Simplifying x2 + -1.5 = 1.204159458 Reorder the terms: -1.5 + x2 = 1.204159458 Solving -1.5 + x2 = 1.204159458 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x2 = 1.204159458 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x2 = 1.204159458 + 1.5 x2 = 1.204159458 + 1.5 Combine like terms: 1.204159458 + 1.5 = 2.704159458 x2 = 2.704159458 Simplifying x2 = 2.704159458 Take the square root of each side: x = {-1.644432868, 1.644432868}

Subproblem 2

x2 + -1.5 = -1.204159458 Simplifying x2 + -1.5 = -1.204159458 Reorder the terms: -1.5 + x2 = -1.204159458 Solving -1.5 + x2 = -1.204159458 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x2 = -1.204159458 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x2 = -1.204159458 + 1.5 x2 = -1.204159458 + 1.5 Combine like terms: -1.204159458 + 1.5 = 0.295840542 x2 = 0.295840542 Simplifying x2 = 0.295840542 Take the square root of each side: x = {-0.543912256, 0.543912256}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.644432868, 1.644432868, -0.543912256, 0.543912256}

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