y^4-18y^2+73=0

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Solution for y^4-18y^2+73=0 equation:


Simplifying
y4 + -18y2 + 73 = 0

Reorder the terms:
73 + -18y2 + y4 = 0

Solving
73 + -18y2 + y4 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-73' to each side of the equation.
73 + -18y2 + -73 + y4 = 0 + -73

Reorder the terms:
73 + -73 + -18y2 + y4 = 0 + -73

Combine like terms: 73 + -73 = 0
0 + -18y2 + y4 = 0 + -73
-18y2 + y4 = 0 + -73

Combine like terms: 0 + -73 = -73
-18y2 + y4 = -73

The y term is -18y2.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18y2 + 81 + y4 = -73 + 81

Reorder the terms:
81 + -18y2 + y4 = -73 + 81

Combine like terms: -73 + 81 = 8
81 + -18y2 + y4 = 8

Factor a perfect square on the left side:
(y2 + -9)(y2 + -9) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(y2 + -9) equal to 2.828427125 and -2.828427125.

Subproblem 1

y2 + -9 = 2.828427125 Simplifying y2 + -9 = 2.828427125 Reorder the terms: -9 + y2 = 2.828427125 Solving -9 + y2 = 2.828427125 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + y2 = 2.828427125 + 9 Combine like terms: -9 + 9 = 0 0 + y2 = 2.828427125 + 9 y2 = 2.828427125 + 9 Combine like terms: 2.828427125 + 9 = 11.828427125 y2 = 11.828427125 Simplifying y2 = 11.828427125 Take the square root of each side: y = {-3.439248046, 3.439248046}

Subproblem 2

y2 + -9 = -2.828427125 Simplifying y2 + -9 = -2.828427125 Reorder the terms: -9 + y2 = -2.828427125 Solving -9 + y2 = -2.828427125 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + y2 = -2.828427125 + 9 Combine like terms: -9 + 9 = 0 0 + y2 = -2.828427125 + 9 y2 = -2.828427125 + 9 Combine like terms: -2.828427125 + 9 = 6.171572875 y2 = 6.171572875 Simplifying y2 = 6.171572875 Take the square root of each side: y = {-2.484265057, 2.484265057}

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-3.439248046, 3.439248046, -2.484265057, 2.484265057}

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