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

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


Simplifying
x4 + -9x2 + 9 = 0

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

Solving
9 + -9x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = -9 + 20.25

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

Combine like terms: -9 + 20.25 = 11.25
20.25 + -9x2 + x4 = 11.25

Factor a perfect square on the left side:
(x2 + -4.5)(x2 + -4.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(x2 + -4.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

x2 + -4.5 = 3.354101966 Simplifying x2 + -4.5 = 3.354101966 Reorder the terms: -4.5 + x2 = 3.354101966 Solving -4.5 + x2 = 3.354101966 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 = 3.354101966 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x2 = 3.354101966 + 4.5 x2 = 3.354101966 + 4.5 Combine like terms: 3.354101966 + 4.5 = 7.854101966 x2 = 7.854101966 Simplifying x2 = 7.854101966 Take the square root of each side: x = {-2.802517077, 2.802517077}

Subproblem 2

x2 + -4.5 = -3.354101966 Simplifying x2 + -4.5 = -3.354101966 Reorder the terms: -4.5 + x2 = -3.354101966 Solving -4.5 + x2 = -3.354101966 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 = -3.354101966 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x2 = -3.354101966 + 4.5 x2 = -3.354101966 + 4.5 Combine like terms: -3.354101966 + 4.5 = 1.145898034 x2 = 1.145898034 Simplifying x2 = 1.145898034 Take the square root of each side: x = {-1.070466269, 1.070466269}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.802517077, 2.802517077, -1.070466269, 1.070466269}

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