3y^4+7y^2-54=0

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Solution for 3y^4+7y^2-54=0 equation:


Simplifying
3y4 + 7y2 + -54 = 0

Reorder the terms:
-54 + 7y2 + 3y4 = 0

Solving
-54 + 7y2 + 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'.
-18 + 2.333333333y2 + y4 = 0

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + 2.333333333y2 + 18 + y4 = 0 + 18

Reorder the terms:
-18 + 18 + 2.333333333y2 + y4 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + 2.333333333y2 + y4 = 0 + 18
2.333333333y2 + y4 = 0 + 18

Combine like terms: 0 + 18 = 18
2.333333333y2 + y4 = 18

The y term is 2.333333333y2.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333y2 + 1.361111112 + y4 = 18 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333y2 + y4 = 18 + 1.361111112

Combine like terms: 18 + 1.361111112 = 19.361111112
1.361111112 + 2.333333333y2 + y4 = 19.361111112

Factor a perfect square on the left side:
(y2 + 1.166666667)(y2 + 1.166666667) = 19.361111112

Calculate the square root of the right side: 4.400126261

Break this problem into two subproblems by setting 
(y2 + 1.166666667) equal to 4.400126261 and -4.400126261.

Subproblem 1

y2 + 1.166666667 = 4.400126261 Simplifying y2 + 1.166666667 = 4.400126261 Reorder the terms: 1.166666667 + y2 = 4.400126261 Solving 1.166666667 + y2 = 4.400126261 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + y2 = 4.400126261 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + y2 = 4.400126261 + -1.166666667 y2 = 4.400126261 + -1.166666667 Combine like terms: 4.400126261 + -1.166666667 = 3.233459594 y2 = 3.233459594 Simplifying y2 = 3.233459594 Take the square root of each side: y = {-1.798182303, 1.798182303}

Subproblem 2

y2 + 1.166666667 = -4.400126261 Simplifying y2 + 1.166666667 = -4.400126261 Reorder the terms: 1.166666667 + y2 = -4.400126261 Solving 1.166666667 + y2 = -4.400126261 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + y2 = -4.400126261 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + y2 = -4.400126261 + -1.166666667 y2 = -4.400126261 + -1.166666667 Combine like terms: -4.400126261 + -1.166666667 = -5.566792928 y2 = -5.566792928 Simplifying y2 = -5.566792928 Reorder the terms: 5.566792928 + y2 = -5.566792928 + 5.566792928 Combine like terms: -5.566792928 + 5.566792928 = 0.000000000 5.566792928 + y2 = 0.000000000 The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined.

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