2y^4-7y^2+1=0

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Solution for 2y^4-7y^2+1=0 equation:


Simplifying
2y4 + -7y2 + 1 = 0

Reorder the terms:
1 + -7y2 + 2y4 = 0

Solving
1 + -7y2 + 2y4 = 0

Solving for variable 'y'.

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

Divide each side by '2'.
0.5 + -3.5y2 + y4 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -3.5y2 + -0.5 + y4 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -3.5y2 + y4 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -3.5y2 + y4 = 0 + -0.5
-3.5y2 + y4 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-3.5y2 + y4 = -0.5

The y term is -3.5y2.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5y2 + 3.0625 + y4 = -0.5 + 3.0625

Reorder the terms:
3.0625 + -3.5y2 + y4 = -0.5 + 3.0625

Combine like terms: -0.5 + 3.0625 = 2.5625
3.0625 + -3.5y2 + y4 = 2.5625

Factor a perfect square on the left side:
(y2 + -1.75)(y2 + -1.75) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(y2 + -1.75) equal to 1.600781059 and -1.600781059.

Subproblem 1

y2 + -1.75 = 1.600781059 Simplifying y2 + -1.75 = 1.600781059 Reorder the terms: -1.75 + y2 = 1.600781059 Solving -1.75 + y2 = 1.600781059 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + y2 = 1.600781059 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + y2 = 1.600781059 + 1.75 y2 = 1.600781059 + 1.75 Combine like terms: 1.600781059 + 1.75 = 3.350781059 y2 = 3.350781059 Simplifying y2 = 3.350781059 Take the square root of each side: y = {-1.830513878, 1.830513878}

Subproblem 2

y2 + -1.75 = -1.600781059 Simplifying y2 + -1.75 = -1.600781059 Reorder the terms: -1.75 + y2 = -1.600781059 Solving -1.75 + y2 = -1.600781059 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + y2 = -1.600781059 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + y2 = -1.600781059 + 1.75 y2 = -1.600781059 + 1.75 Combine like terms: -1.600781059 + 1.75 = 0.149218941 y2 = 0.149218941 Simplifying y2 = 0.149218941 Take the square root of each side: y = {-0.386288676, 0.386288676}

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-1.830513878, 1.830513878, -0.386288676, 0.386288676}

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