0=x^4-18x^2+20

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Solution for 0=x^4-18x^2+20 equation:


Simplifying
0 = x4 + -18x2 + 20

Reorder the terms:
0 = 20 + -18x2 + x4

Solving
0 = 20 + -18x2 + x4

Solving for variable 'x'.

Combine like terms: 0 + -20 = -20
-20 + 18x2 + -1x4 = 20 + -18x2 + x4 + -20 + 18x2 + -1x4

Reorder the terms:
-20 + 18x2 + -1x4 = 20 + -20 + -18x2 + 18x2 + x4 + -1x4

Combine like terms: 20 + -20 = 0
-20 + 18x2 + -1x4 = 0 + -18x2 + 18x2 + x4 + -1x4
-20 + 18x2 + -1x4 = -18x2 + 18x2 + x4 + -1x4

Combine like terms: -18x2 + 18x2 = 0
-20 + 18x2 + -1x4 = 0 + x4 + -1x4
-20 + 18x2 + -1x4 = x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-20 + 18x2 + -1x4 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
20 + -18x2 + x4 = 0

Move the constant term to the right:

Add '-20' to each side of the equation.
20 + -18x2 + -20 + x4 = 0 + -20

Reorder the terms:
20 + -20 + -18x2 + x4 = 0 + -20

Combine like terms: 20 + -20 = 0
0 + -18x2 + x4 = 0 + -20
-18x2 + x4 = 0 + -20

Combine like terms: 0 + -20 = -20
-18x2 + x4 = -20

The x term is -18x2.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18x2 + 81 + x4 = -20 + 81

Reorder the terms:
81 + -18x2 + x4 = -20 + 81

Combine like terms: -20 + 81 = 61
81 + -18x2 + x4 = 61

Factor a perfect square on the left side:
(x2 + -9)(x2 + -9) = 61

Calculate the square root of the right side: 7.810249676

Break this problem into two subproblems by setting 
(x2 + -9) equal to 7.810249676 and -7.810249676.

Subproblem 1

x2 + -9 = 7.810249676 Simplifying x2 + -9 = 7.810249676 Reorder the terms: -9 + x2 = 7.810249676 Solving -9 + x2 = 7.810249676 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x2 = 7.810249676 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = 7.810249676 + 9 x2 = 7.810249676 + 9 Combine like terms: 7.810249676 + 9 = 16.810249676 x2 = 16.810249676 Simplifying x2 = 16.810249676 Take the square root of each side: x = {-4.100030448, 4.100030448}

Subproblem 2

x2 + -9 = -7.810249676 Simplifying x2 + -9 = -7.810249676 Reorder the terms: -9 + x2 = -7.810249676 Solving -9 + x2 = -7.810249676 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x2 = -7.810249676 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = -7.810249676 + 9 x2 = -7.810249676 + 9 Combine like terms: -7.810249676 + 9 = 1.189750324 x2 = 1.189750324 Simplifying x2 = 1.189750324 Take the square root of each side: x = {-1.090756767, 1.090756767}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.100030448, 4.100030448, -1.090756767, 1.090756767}

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