x^4-19x^2+80=0

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Solution for x^4-19x^2+80=0 equation:


Simplifying
x4 + -19x2 + 80 = 0

Reorder the terms:
80 + -19x2 + x4 = 0

Solving
80 + -19x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-80' to each side of the equation.
80 + -19x2 + -80 + x4 = 0 + -80

Reorder the terms:
80 + -80 + -19x2 + x4 = 0 + -80

Combine like terms: 80 + -80 = 0
0 + -19x2 + x4 = 0 + -80
-19x2 + x4 = 0 + -80

Combine like terms: 0 + -80 = -80
-19x2 + x4 = -80

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 = -80 + 90.25

Reorder the terms:
90.25 + -19x2 + x4 = -80 + 90.25

Combine like terms: -80 + 90.25 = 10.25
90.25 + -19x2 + x4 = 10.25

Factor a perfect square on the left side:
(x2 + -9.5)(x2 + -9.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(x2 + -9.5) equal to 3.201562119 and -3.201562119.

Subproblem 1

x2 + -9.5 = 3.201562119 Simplifying x2 + -9.5 = 3.201562119 Reorder the terms: -9.5 + x2 = 3.201562119 Solving -9.5 + x2 = 3.201562119 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 = 3.201562119 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + x2 = 3.201562119 + 9.5 x2 = 3.201562119 + 9.5 Combine like terms: 3.201562119 + 9.5 = 12.701562119 x2 = 12.701562119 Simplifying x2 = 12.701562119 Take the square root of each side: x = {-3.5639251, 3.5639251}

Subproblem 2

x2 + -9.5 = -3.201562119 Simplifying x2 + -9.5 = -3.201562119 Reorder the terms: -9.5 + x2 = -3.201562119 Solving -9.5 + x2 = -3.201562119 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 = -3.201562119 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + x2 = -3.201562119 + 9.5 x2 = -3.201562119 + 9.5 Combine like terms: -3.201562119 + 9.5 = 6.298437881 x2 = 6.298437881 Simplifying x2 = 6.298437881 Take the square root of each side: x = {-2.509668879, 2.509668879}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.5639251, 3.5639251, -2.509668879, 2.509668879}

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