x^4-24x^2+1=0

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Solution for x^4-24x^2+1=0 equation:


Simplifying
x4 + -24x2 + 1 = 0

Reorder the terms:
1 + -24x2 + x4 = 0

Solving
1 + -24x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -24x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -24x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -24x2 + x4 = 0 + -1
-24x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-24x2 + x4 = -1

The x term is -24x2.  Take half its coefficient (-12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
-24x2 + 144 + x4 = -1 + 144

Reorder the terms:
144 + -24x2 + x4 = -1 + 144

Combine like terms: -1 + 144 = 143
144 + -24x2 + x4 = 143

Factor a perfect square on the left side:
(x2 + -12)(x2 + -12) = 143

Calculate the square root of the right side: 11.958260743

Break this problem into two subproblems by setting 
(x2 + -12) equal to 11.958260743 and -11.958260743.

Subproblem 1

x2 + -12 = 11.958260743 Simplifying x2 + -12 = 11.958260743 Reorder the terms: -12 + x2 = 11.958260743 Solving -12 + x2 = 11.958260743 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + x2 = 11.958260743 + 12 Combine like terms: -12 + 12 = 0 0 + x2 = 11.958260743 + 12 x2 = 11.958260743 + 12 Combine like terms: 11.958260743 + 12 = 23.958260743 x2 = 23.958260743 Simplifying x2 = 23.958260743 Take the square root of each side: x = {-4.894717637, 4.894717637}

Subproblem 2

x2 + -12 = -11.958260743 Simplifying x2 + -12 = -11.958260743 Reorder the terms: -12 + x2 = -11.958260743 Solving -12 + x2 = -11.958260743 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + x2 = -11.958260743 + 12 Combine like terms: -12 + 12 = 0 0 + x2 = -11.958260743 + 12 x2 = -11.958260743 + 12 Combine like terms: -11.958260743 + 12 = 0.041739257 x2 = 0.041739257 Simplifying x2 = 0.041739257 Take the square root of each side: x = {-0.204301877, 0.204301877}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.894717637, 4.894717637, -0.204301877, 0.204301877}

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