x^4-4.25x^2+1=0

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Solution for x^4-4.25x^2+1=0 equation:


Simplifying
x4 + -4.25x2 + 1 = 0

Reorder the terms:
1 + -4.25x2 + x4 = 0

Solving
1 + -4.25x2 + 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 + -4.25x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -4.25x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -4.25x2 + x4 = 0 + -1
-4.25x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-4.25x2 + x4 = -1

The x term is -4.25x2.  Take half its coefficient (-2.125).
Square it (4.515625) and add it to both sides.

Add '4.515625' to each side of the equation.
-4.25x2 + 4.515625 + x4 = -1 + 4.515625

Reorder the terms:
4.515625 + -4.25x2 + x4 = -1 + 4.515625

Combine like terms: -1 + 4.515625 = 3.515625
4.515625 + -4.25x2 + x4 = 3.515625

Factor a perfect square on the left side:
(x2 + -2.125)(x2 + -2.125) = 3.515625

Calculate the square root of the right side: 1.875

Break this problem into two subproblems by setting 
(x2 + -2.125) equal to 1.875 and -1.875.

Subproblem 1

x2 + -2.125 = 1.875 Simplifying x2 + -2.125 = 1.875 Reorder the terms: -2.125 + x2 = 1.875 Solving -2.125 + x2 = 1.875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.125' to each side of the equation. -2.125 + 2.125 + x2 = 1.875 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + x2 = 1.875 + 2.125 x2 = 1.875 + 2.125 Combine like terms: 1.875 + 2.125 = 4 x2 = 4 Simplifying x2 = 4 Take the square root of each side: x = {-2, 2}

Subproblem 2

x2 + -2.125 = -1.875 Simplifying x2 + -2.125 = -1.875 Reorder the terms: -2.125 + x2 = -1.875 Solving -2.125 + x2 = -1.875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.125' to each side of the equation. -2.125 + 2.125 + x2 = -1.875 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + x2 = -1.875 + 2.125 x2 = -1.875 + 2.125 Combine like terms: -1.875 + 2.125 = 0.25 x2 = 0.25 Simplifying x2 = 0.25 Take the square root of each side: x = {-0.5, 0.5}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2, 2, -0.5, 0.5}

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