x^4-9x^2+6=0

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Solution for x^4-9x^2+6=0 equation:


Simplifying
x4 + -9x2 + 6 = 0

Reorder the terms:
6 + -9x2 + x4 = 0

Solving
6 + -9x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -9x2 + -6 + x4 = 0 + -6

Reorder the terms:
6 + -6 + -9x2 + x4 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -9x2 + x4 = 0 + -6
-9x2 + x4 = 0 + -6

Combine like terms: 0 + -6 = -6
-9x2 + x4 = -6

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 = -6 + 20.25

Reorder the terms:
20.25 + -9x2 + x4 = -6 + 20.25

Combine like terms: -6 + 20.25 = 14.25
20.25 + -9x2 + x4 = 14.25

Factor a perfect square on the left side:
(x2 + -4.5)(x2 + -4.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(x2 + -4.5) equal to 3.774917218 and -3.774917218.

Subproblem 1

x2 + -4.5 = 3.774917218 Simplifying x2 + -4.5 = 3.774917218 Reorder the terms: -4.5 + x2 = 3.774917218 Solving -4.5 + x2 = 3.774917218 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 = 3.774917218 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x2 = 3.774917218 + 4.5 x2 = 3.774917218 + 4.5 Combine like terms: 3.774917218 + 4.5 = 8.274917218 x2 = 8.274917218 Simplifying x2 = 8.274917218 Take the square root of each side: x = {-2.876615584, 2.876615584}

Subproblem 2

x2 + -4.5 = -3.774917218 Simplifying x2 + -4.5 = -3.774917218 Reorder the terms: -4.5 + x2 = -3.774917218 Solving -4.5 + x2 = -3.774917218 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 = -3.774917218 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x2 = -3.774917218 + 4.5 x2 = -3.774917218 + 4.5 Combine like terms: -3.774917218 + 4.5 = 0.725082782 x2 = 0.725082782 Simplifying x2 = 0.725082782 Take the square root of each side: x = {-0.851517928, 0.851517928}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.876615584, 2.876615584, -0.851517928, 0.851517928}

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