x^4-19x^2+81=0

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Solution for x^4-19x^2+81=0 equation:


Simplifying
x4 + -19x2 + 81 = 0

Reorder the terms:
81 + -19x2 + x4 = 0

Solving
81 + -19x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-81' to each side of the equation.
81 + -19x2 + -81 + x4 = 0 + -81

Reorder the terms:
81 + -81 + -19x2 + x4 = 0 + -81

Combine like terms: 81 + -81 = 0
0 + -19x2 + x4 = 0 + -81
-19x2 + x4 = 0 + -81

Combine like terms: 0 + -81 = -81
-19x2 + x4 = -81

The x term is -19x2.  Take half its coefficient (-9.5).
Square it (90.25) and add it to both sides.

Add '90.25' to each side of the equation.
-19x2 + 90.25 + x4 = -81 + 90.25

Reorder the terms:
90.25 + -19x2 + x4 = -81 + 90.25

Combine like terms: -81 + 90.25 = 9.25
90.25 + -19x2 + x4 = 9.25

Factor a perfect square on the left side:
(x2 + -9.5)(x2 + -9.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(x2 + -9.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

x2 + -9.5 = 3.041381265 Simplifying x2 + -9.5 = 3.041381265 Reorder the terms: -9.5 + x2 = 3.041381265 Solving -9.5 + x2 = 3.041381265 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + x2 = 3.041381265 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + x2 = 3.041381265 + 9.5 x2 = 3.041381265 + 9.5 Combine like terms: 3.041381265 + 9.5 = 12.541381265 x2 = 12.541381265 Simplifying x2 = 12.541381265 Take the square root of each side: x = {-3.541381265, 3.541381265}

Subproblem 2

x2 + -9.5 = -3.041381265 Simplifying x2 + -9.5 = -3.041381265 Reorder the terms: -9.5 + x2 = -3.041381265 Solving -9.5 + x2 = -3.041381265 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + x2 = -3.041381265 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + x2 = -3.041381265 + 9.5 x2 = -3.041381265 + 9.5 Combine like terms: -3.041381265 + 9.5 = 6.458618735 x2 = 6.458618735 Simplifying x2 = 6.458618735 Take the square root of each side: x = {-2.541381265, 2.541381265}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.541381265, 3.541381265, -2.541381265, 2.541381265}

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