x^4-22x^2+49=0

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Solution for x^4-22x^2+49=0 equation:


Simplifying
x4 + -22x2 + 49 = 0

Reorder the terms:
49 + -22x2 + x4 = 0

Solving
49 + -22x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-49' to each side of the equation.
49 + -22x2 + -49 + x4 = 0 + -49

Reorder the terms:
49 + -49 + -22x2 + x4 = 0 + -49

Combine like terms: 49 + -49 = 0
0 + -22x2 + x4 = 0 + -49
-22x2 + x4 = 0 + -49

Combine like terms: 0 + -49 = -49
-22x2 + x4 = -49

The x term is -22x2.  Take half its coefficient (-11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
-22x2 + 121 + x4 = -49 + 121

Reorder the terms:
121 + -22x2 + x4 = -49 + 121

Combine like terms: -49 + 121 = 72
121 + -22x2 + x4 = 72

Factor a perfect square on the left side:
(x2 + -11)(x2 + -11) = 72

Calculate the square root of the right side: 8.485281374

Break this problem into two subproblems by setting 
(x2 + -11) equal to 8.485281374 and -8.485281374.

Subproblem 1

x2 + -11 = 8.485281374 Simplifying x2 + -11 = 8.485281374 Reorder the terms: -11 + x2 = 8.485281374 Solving -11 + x2 = 8.485281374 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + x2 = 8.485281374 + 11 Combine like terms: -11 + 11 = 0 0 + x2 = 8.485281374 + 11 x2 = 8.485281374 + 11 Combine like terms: 8.485281374 + 11 = 19.485281374 x2 = 19.485281374 Simplifying x2 = 19.485281374 Take the square root of each side: x = {-4.414213562, 4.414213562}

Subproblem 2

x2 + -11 = -8.485281374 Simplifying x2 + -11 = -8.485281374 Reorder the terms: -11 + x2 = -8.485281374 Solving -11 + x2 = -8.485281374 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + x2 = -8.485281374 + 11 Combine like terms: -11 + 11 = 0 0 + x2 = -8.485281374 + 11 x2 = -8.485281374 + 11 Combine like terms: -8.485281374 + 11 = 2.514718626 x2 = 2.514718626 Simplifying x2 = 2.514718626 Take the square root of each side: x = {-1.585786438, 1.585786438}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.414213562, 4.414213562, -1.585786438, 1.585786438}

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