x^4-27x^2+100=0

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Solution for x^4-27x^2+100=0 equation:


Simplifying
x4 + -27x2 + 100 = 0

Reorder the terms:
100 + -27x2 + x4 = 0

Solving
100 + -27x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-100' to each side of the equation.
100 + -27x2 + -100 + x4 = 0 + -100

Reorder the terms:
100 + -100 + -27x2 + x4 = 0 + -100

Combine like terms: 100 + -100 = 0
0 + -27x2 + x4 = 0 + -100
-27x2 + x4 = 0 + -100

Combine like terms: 0 + -100 = -100
-27x2 + x4 = -100

The x term is -27x2.  Take half its coefficient (-13.5).
Square it (182.25) and add it to both sides.

Add '182.25' to each side of the equation.
-27x2 + 182.25 + x4 = -100 + 182.25

Reorder the terms:
182.25 + -27x2 + x4 = -100 + 182.25

Combine like terms: -100 + 182.25 = 82.25
182.25 + -27x2 + x4 = 82.25

Factor a perfect square on the left side:
(x2 + -13.5)(x2 + -13.5) = 82.25

Calculate the square root of the right side: 9.069178574

Break this problem into two subproblems by setting 
(x2 + -13.5) equal to 9.069178574 and -9.069178574.

Subproblem 1

x2 + -13.5 = 9.069178574 Simplifying x2 + -13.5 = 9.069178574 Reorder the terms: -13.5 + x2 = 9.069178574 Solving -13.5 + x2 = 9.069178574 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + x2 = 9.069178574 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + x2 = 9.069178574 + 13.5 x2 = 9.069178574 + 13.5 Combine like terms: 9.069178574 + 13.5 = 22.569178574 x2 = 22.569178574 Simplifying x2 = 22.569178574 Take the square root of each side: x = {-4.750702956, 4.750702956}

Subproblem 2

x2 + -13.5 = -9.069178574 Simplifying x2 + -13.5 = -9.069178574 Reorder the terms: -13.5 + x2 = -9.069178574 Solving -13.5 + x2 = -9.069178574 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + x2 = -9.069178574 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + x2 = -9.069178574 + 13.5 x2 = -9.069178574 + 13.5 Combine like terms: -9.069178574 + 13.5 = 4.430821426 x2 = 4.430821426 Simplifying x2 = 4.430821426 Take the square root of each side: x = {-2.104951645, 2.104951645}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.750702956, 4.750702956, -2.104951645, 2.104951645}

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