X^4-26X^2+60=0

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Solution for X^4-26X^2+60=0 equation:


Simplifying
X4 + -26X2 + 60 = 0

Reorder the terms:
60 + -26X2 + X4 = 0

Solving
60 + -26X2 + X4 = 0

Solving for variable 'X'.

Begin completing the square.

Move the constant term to the right:

Add '-60' to each side of the equation.
60 + -26X2 + -60 + X4 = 0 + -60

Reorder the terms:
60 + -60 + -26X2 + X4 = 0 + -60

Combine like terms: 60 + -60 = 0
0 + -26X2 + X4 = 0 + -60
-26X2 + X4 = 0 + -60

Combine like terms: 0 + -60 = -60
-26X2 + X4 = -60

The X term is -26X2.  Take half its coefficient (-13).
Square it (169) and add it to both sides.

Add '169' to each side of the equation.
-26X2 + 169 + X4 = -60 + 169

Reorder the terms:
169 + -26X2 + X4 = -60 + 169

Combine like terms: -60 + 169 = 109
169 + -26X2 + X4 = 109

Factor a perfect square on the left side:
(X2 + -13)(X2 + -13) = 109

Calculate the square root of the right side: 10.440306509

Break this problem into two subproblems by setting 
(X2 + -13) equal to 10.440306509 and -10.440306509.

Subproblem 1

X2 + -13 = 10.440306509 Simplifying X2 + -13 = 10.440306509 Reorder the terms: -13 + X2 = 10.440306509 Solving -13 + X2 = 10.440306509 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '13' to each side of the equation. -13 + 13 + X2 = 10.440306509 + 13 Combine like terms: -13 + 13 = 0 0 + X2 = 10.440306509 + 13 X2 = 10.440306509 + 13 Combine like terms: 10.440306509 + 13 = 23.440306509 X2 = 23.440306509 Simplifying X2 = 23.440306509 Take the square root of each side: X = {-4.841519029, 4.841519029}

Subproblem 2

X2 + -13 = -10.440306509 Simplifying X2 + -13 = -10.440306509 Reorder the terms: -13 + X2 = -10.440306509 Solving -13 + X2 = -10.440306509 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '13' to each side of the equation. -13 + 13 + X2 = -10.440306509 + 13 Combine like terms: -13 + 13 = 0 0 + X2 = -10.440306509 + 13 X2 = -10.440306509 + 13 Combine like terms: -10.440306509 + 13 = 2.559693491 X2 = 2.559693491 Simplifying X2 = 2.559693491 Take the square root of each side: X = {-1.599904213, 1.599904213}

Solution

The solution to the problem is based on the solutions from the subproblems. X = {-4.841519029, 4.841519029, -1.599904213, 1.599904213}

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