w^4-22w^2+13=0

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Solution for w^4-22w^2+13=0 equation:


Simplifying
w4 + -22w2 + 13 = 0

Reorder the terms:
13 + -22w2 + w4 = 0

Solving
13 + -22w2 + w4 = 0

Solving for variable 'w'.

Begin completing the square.

Move the constant term to the right:

Add '-13' to each side of the equation.
13 + -22w2 + -13 + w4 = 0 + -13

Reorder the terms:
13 + -13 + -22w2 + w4 = 0 + -13

Combine like terms: 13 + -13 = 0
0 + -22w2 + w4 = 0 + -13
-22w2 + w4 = 0 + -13

Combine like terms: 0 + -13 = -13
-22w2 + w4 = -13

The w term is -22w2.  Take half its coefficient (-11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
-22w2 + 121 + w4 = -13 + 121

Reorder the terms:
121 + -22w2 + w4 = -13 + 121

Combine like terms: -13 + 121 = 108
121 + -22w2 + w4 = 108

Factor a perfect square on the left side:
(w2 + -11)(w2 + -11) = 108

Calculate the square root of the right side: 10.392304845

Break this problem into two subproblems by setting 
(w2 + -11) equal to 10.392304845 and -10.392304845.

Subproblem 1

w2 + -11 = 10.392304845 Simplifying w2 + -11 = 10.392304845 Reorder the terms: -11 + w2 = 10.392304845 Solving -11 + w2 = 10.392304845 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + w2 = 10.392304845 + 11 Combine like terms: -11 + 11 = 0 0 + w2 = 10.392304845 + 11 w2 = 10.392304845 + 11 Combine like terms: 10.392304845 + 11 = 21.392304845 w2 = 21.392304845 Simplifying w2 = 21.392304845 Take the square root of each side: w = {-4.625181601, 4.625181601}

Subproblem 2

w2 + -11 = -10.392304845 Simplifying w2 + -11 = -10.392304845 Reorder the terms: -11 + w2 = -10.392304845 Solving -11 + w2 = -10.392304845 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + w2 = -10.392304845 + 11 Combine like terms: -11 + 11 = 0 0 + w2 = -10.392304845 + 11 w2 = -10.392304845 + 11 Combine like terms: -10.392304845 + 11 = 0.607695155 w2 = 0.607695155 Simplifying w2 = 0.607695155 Take the square root of each side: w = {-0.779548045, 0.779548045}

Solution

The solution to the problem is based on the solutions from the subproblems. w = {-4.625181601, 4.625181601, -0.779548045, 0.779548045}

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