x^4-27x^2+144=0

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Solution for x^4-27x^2+144=0 equation:


Simplifying
x4 + -27x2 + 144 = 0

Reorder the terms:
144 + -27x2 + x4 = 0

Solving
144 + -27x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-144' to each side of the equation.
144 + -27x2 + -144 + x4 = 0 + -144

Reorder the terms:
144 + -144 + -27x2 + x4 = 0 + -144

Combine like terms: 144 + -144 = 0
0 + -27x2 + x4 = 0 + -144
-27x2 + x4 = 0 + -144

Combine like terms: 0 + -144 = -144
-27x2 + x4 = -144

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 = -144 + 182.25

Reorder the terms:
182.25 + -27x2 + x4 = -144 + 182.25

Combine like terms: -144 + 182.25 = 38.25
182.25 + -27x2 + x4 = 38.25

Factor a perfect square on the left side:
(x2 + -13.5)(x2 + -13.5) = 38.25

Calculate the square root of the right side: 6.184658438

Break this problem into two subproblems by setting 
(x2 + -13.5) equal to 6.184658438 and -6.184658438.

Subproblem 1

x2 + -13.5 = 6.184658438 Simplifying x2 + -13.5 = 6.184658438 Reorder the terms: -13.5 + x2 = 6.184658438 Solving -13.5 + x2 = 6.184658438 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 = 6.184658438 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + x2 = 6.184658438 + 13.5 x2 = 6.184658438 + 13.5 Combine like terms: 6.184658438 + 13.5 = 19.684658438 x2 = 19.684658438 Simplifying x2 = 19.684658438 Take the square root of each side: x = {-4.436739618, 4.436739618}

Subproblem 2

x2 + -13.5 = -6.184658438 Simplifying x2 + -13.5 = -6.184658438 Reorder the terms: -13.5 + x2 = -6.184658438 Solving -13.5 + x2 = -6.184658438 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 = -6.184658438 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + x2 = -6.184658438 + 13.5 x2 = -6.184658438 + 13.5 Combine like terms: -6.184658438 + 13.5 = 7.315341562 x2 = 7.315341562 Simplifying x2 = 7.315341562 Take the square root of each side: x = {-2.704688811, 2.704688811}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.436739618, 4.436739618, -2.704688811, 2.704688811}

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