x^4-9x^2+1=0

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Solution for x^4-9x^2+1=0 equation:


Simplifying
x4 + -9x2 + 1 = 0

Reorder the terms:
1 + -9x2 + x4 = 0

Solving
1 + -9x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -9x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -9x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -9x2 + x4 = 0 + -1
-9x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-9x2 + x4 = -1

The x term is -9x2.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9x2 + 20.25 + x4 = -1 + 20.25

Reorder the terms:
20.25 + -9x2 + x4 = -1 + 20.25

Combine like terms: -1 + 20.25 = 19.25
20.25 + -9x2 + x4 = 19.25

Factor a perfect square on the left side:
(x2 + -4.5)(x2 + -4.5) = 19.25

Calculate the square root of the right side: 4.387482194

Break this problem into two subproblems by setting 
(x2 + -4.5) equal to 4.387482194 and -4.387482194.

Subproblem 1

x2 + -4.5 = 4.387482194 Simplifying x2 + -4.5 = 4.387482194 Reorder the terms: -4.5 + x2 = 4.387482194 Solving -4.5 + x2 = 4.387482194 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x2 = 4.387482194 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x2 = 4.387482194 + 4.5 x2 = 4.387482194 + 4.5 Combine like terms: 4.387482194 + 4.5 = 8.887482194 x2 = 8.887482194 Simplifying x2 = 8.887482194 Take the square root of each side: x = {-2.981188051, 2.981188051}

Subproblem 2

x2 + -4.5 = -4.387482194 Simplifying x2 + -4.5 = -4.387482194 Reorder the terms: -4.5 + x2 = -4.387482194 Solving -4.5 + x2 = -4.387482194 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x2 = -4.387482194 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x2 = -4.387482194 + 4.5 x2 = -4.387482194 + 4.5 Combine like terms: -4.387482194 + 4.5 = 0.112517806 x2 = 0.112517806 Simplifying x2 = 0.112517806 Take the square root of each side: x = {-0.335436739, 0.335436739}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.981188051, 2.981188051, -0.335436739, 0.335436739}

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