n^4-2n^2-84=0

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Solution for n^4-2n^2-84=0 equation:


Simplifying
n4 + -2n2 + -84 = 0

Reorder the terms:
-84 + -2n2 + n4 = 0

Solving
-84 + -2n2 + n4 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '84' to each side of the equation.
-84 + -2n2 + 84 + n4 = 0 + 84

Reorder the terms:
-84 + 84 + -2n2 + n4 = 0 + 84

Combine like terms: -84 + 84 = 0
0 + -2n2 + n4 = 0 + 84
-2n2 + n4 = 0 + 84

Combine like terms: 0 + 84 = 84
-2n2 + n4 = 84

The n term is -2n2.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n2 + 1 + n4 = 84 + 1

Reorder the terms:
1 + -2n2 + n4 = 84 + 1

Combine like terms: 84 + 1 = 85
1 + -2n2 + n4 = 85

Factor a perfect square on the left side:
(n2 + -1)(n2 + -1) = 85

Calculate the square root of the right side: 9.219544457

Break this problem into two subproblems by setting 
(n2 + -1) equal to 9.219544457 and -9.219544457.

Subproblem 1

n2 + -1 = 9.219544457 Simplifying n2 + -1 = 9.219544457 Reorder the terms: -1 + n2 = 9.219544457 Solving -1 + n2 = 9.219544457 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n2 = 9.219544457 + 1 Combine like terms: -1 + 1 = 0 0 + n2 = 9.219544457 + 1 n2 = 9.219544457 + 1 Combine like terms: 9.219544457 + 1 = 10.219544457 n2 = 10.219544457 Simplifying n2 = 10.219544457 Take the square root of each side: n = {-3.196802224, 3.196802224}

Subproblem 2

n2 + -1 = -9.219544457 Simplifying n2 + -1 = -9.219544457 Reorder the terms: -1 + n2 = -9.219544457 Solving -1 + n2 = -9.219544457 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n2 = -9.219544457 + 1 Combine like terms: -1 + 1 = 0 0 + n2 = -9.219544457 + 1 n2 = -9.219544457 + 1 Combine like terms: -9.219544457 + 1 = -8.219544457 n2 = -8.219544457 Simplifying n2 = -8.219544457 Reorder the terms: 8.219544457 + n2 = -8.219544457 + 8.219544457 Combine like terms: -8.219544457 + 8.219544457 = 0.000000000 8.219544457 + n2 = 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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