y^4+12y^2-24=0

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Solution for y^4+12y^2-24=0 equation:


Simplifying
y4 + 12y2 + -24 = 0

Reorder the terms:
-24 + 12y2 + y4 = 0

Solving
-24 + 12y2 + y4 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + 12y2 + 24 + y4 = 0 + 24

Reorder the terms:
-24 + 24 + 12y2 + y4 = 0 + 24

Combine like terms: -24 + 24 = 0
0 + 12y2 + y4 = 0 + 24
12y2 + y4 = 0 + 24

Combine like terms: 0 + 24 = 24
12y2 + y4 = 24

The y term is 12y2.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12y2 + 36 + y4 = 24 + 36

Reorder the terms:
36 + 12y2 + y4 = 24 + 36

Combine like terms: 24 + 36 = 60
36 + 12y2 + y4 = 60

Factor a perfect square on the left side:
(y2 + 6)(y2 + 6) = 60

Calculate the square root of the right side: 7.745966692

Break this problem into two subproblems by setting 
(y2 + 6) equal to 7.745966692 and -7.745966692.

Subproblem 1

y2 + 6 = 7.745966692 Simplifying y2 + 6 = 7.745966692 Reorder the terms: 6 + y2 = 7.745966692 Solving 6 + y2 = 7.745966692 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + y2 = 7.745966692 + -6 Combine like terms: 6 + -6 = 0 0 + y2 = 7.745966692 + -6 y2 = 7.745966692 + -6 Combine like terms: 7.745966692 + -6 = 1.745966692 y2 = 1.745966692 Simplifying y2 = 1.745966692 Take the square root of each side: y = {-1.321350329, 1.321350329}

Subproblem 2

y2 + 6 = -7.745966692 Simplifying y2 + 6 = -7.745966692 Reorder the terms: 6 + y2 = -7.745966692 Solving 6 + y2 = -7.745966692 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + y2 = -7.745966692 + -6 Combine like terms: 6 + -6 = 0 0 + y2 = -7.745966692 + -6 y2 = -7.745966692 + -6 Combine like terms: -7.745966692 + -6 = -13.745966692 y2 = -13.745966692 Simplifying y2 = -13.745966692 Reorder the terms: 13.745966692 + y2 = -13.745966692 + 13.745966692 Combine like terms: -13.745966692 + 13.745966692 = 0.000000000 13.745966692 + 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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