x^4-18x^2+67=0

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Solution for x^4-18x^2+67=0 equation:


Simplifying
x4 + -18x2 + 67 = 0

Reorder the terms:
67 + -18x2 + x4 = 0

Solving
67 + -18x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-67' to each side of the equation.
67 + -18x2 + -67 + x4 = 0 + -67

Reorder the terms:
67 + -67 + -18x2 + x4 = 0 + -67

Combine like terms: 67 + -67 = 0
0 + -18x2 + x4 = 0 + -67
-18x2 + x4 = 0 + -67

Combine like terms: 0 + -67 = -67
-18x2 + x4 = -67

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 = -67 + 81

Reorder the terms:
81 + -18x2 + x4 = -67 + 81

Combine like terms: -67 + 81 = 14
81 + -18x2 + x4 = 14

Factor a perfect square on the left side:
(x2 + -9)(x2 + -9) = 14

Calculate the square root of the right side: 3.741657387

Break this problem into two subproblems by setting 
(x2 + -9) equal to 3.741657387 and -3.741657387.

Subproblem 1

x2 + -9 = 3.741657387 Simplifying x2 + -9 = 3.741657387 Reorder the terms: -9 + x2 = 3.741657387 Solving -9 + x2 = 3.741657387 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 = 3.741657387 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = 3.741657387 + 9 x2 = 3.741657387 + 9 Combine like terms: 3.741657387 + 9 = 12.741657387 x2 = 12.741657387 Simplifying x2 = 12.741657387 Take the square root of each side: x = {-3.569545824, 3.569545824}

Subproblem 2

x2 + -9 = -3.741657387 Simplifying x2 + -9 = -3.741657387 Reorder the terms: -9 + x2 = -3.741657387 Solving -9 + x2 = -3.741657387 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 = -3.741657387 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = -3.741657387 + 9 x2 = -3.741657387 + 9 Combine like terms: -3.741657387 + 9 = 5.258342613 x2 = 5.258342613 Simplifying x2 = 5.258342613 Take the square root of each side: x = {-2.293107632, 2.293107632}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.569545824, 3.569545824, -2.293107632, 2.293107632}

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