3y^4-5y^2+1=0

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Solution for 3y^4-5y^2+1=0 equation:


Simplifying
3y4 + -5y2 + 1 = 0

Reorder the terms:
1 + -5y2 + 3y4 = 0

Solving
1 + -5y2 + 3y4 = 0

Solving for variable 'y'.

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

Divide each side by '3'.
0.3333333333 + -1.666666667y2 + y4 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -1.666666667y2 + -0.3333333333 + y4 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -1.666666667y2 + y4 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -1.666666667y2 + y4 = 0 + -0.3333333333
-1.666666667y2 + y4 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-1.666666667y2 + y4 = -0.3333333333

The y term is -1.666666667y2.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667y2 + 0.6944444447 + y4 = -0.3333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667y2 + y4 = -0.3333333333 + 0.6944444447

Combine like terms: -0.3333333333 + 0.6944444447 = 0.3611111114
0.6944444447 + -1.666666667y2 + y4 = 0.3611111114

Factor a perfect square on the left side:
(y2 + -0.8333333335)(y2 + -0.8333333335) = 0.3611111114

Calculate the square root of the right side: 0.600925213

Break this problem into two subproblems by setting 
(y2 + -0.8333333335) equal to 0.600925213 and -0.600925213.

Subproblem 1

y2 + -0.8333333335 = 0.600925213 Simplifying y2 + -0.8333333335 = 0.600925213 Reorder the terms: -0.8333333335 + y2 = 0.600925213 Solving -0.8333333335 + y2 = 0.600925213 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y2 = 0.600925213 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y2 = 0.600925213 + 0.8333333335 y2 = 0.600925213 + 0.8333333335 Combine like terms: 0.600925213 + 0.8333333335 = 1.4342585465 y2 = 1.4342585465 Simplifying y2 = 1.4342585465 Take the square root of each side: y = {-1.197605338, 1.197605338}

Subproblem 2

y2 + -0.8333333335 = -0.600925213 Simplifying y2 + -0.8333333335 = -0.600925213 Reorder the terms: -0.8333333335 + y2 = -0.600925213 Solving -0.8333333335 + y2 = -0.600925213 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y2 = -0.600925213 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y2 = -0.600925213 + 0.8333333335 y2 = -0.600925213 + 0.8333333335 Combine like terms: -0.600925213 + 0.8333333335 = 0.2324081205 y2 = 0.2324081205 Simplifying y2 = 0.2324081205 Take the square root of each side: y = {-0.482087254, 0.482087254}

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-1.197605338, 1.197605338, -0.482087254, 0.482087254}

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