x^4-16.25x^2=-4

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Solution for x^4-16.25x^2=-4 equation:


Simplifying
x4 + -16.25x2 = -4

Reorder the terms:
-16.25x2 + x4 = -4

Solving
-16.25x2 + x4 = -4

Solving for variable 'x'.

Reorder the terms:
4 + -16.25x2 + x4 = -4 + 4

Combine like terms: -4 + 4 = 0
4 + -16.25x2 + x4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + -16.25x2 + -4 + x4 = 0 + -4

Reorder the terms:
4 + -4 + -16.25x2 + x4 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -16.25x2 + x4 = 0 + -4
-16.25x2 + x4 = 0 + -4

Combine like terms: 0 + -4 = -4
-16.25x2 + x4 = -4

The x term is -16.25x2.  Take half its coefficient (-8.125).
Square it (66.015625) and add it to both sides.

Add '66.015625' to each side of the equation.
-16.25x2 + 66.015625 + x4 = -4 + 66.015625

Reorder the terms:
66.015625 + -16.25x2 + x4 = -4 + 66.015625

Combine like terms: -4 + 66.015625 = 62.015625
66.015625 + -16.25x2 + x4 = 62.015625

Factor a perfect square on the left side:
(x2 + -8.125)(x2 + -8.125) = 62.015625

Calculate the square root of the right side: 7.875

Break this problem into two subproblems by setting 
(x2 + -8.125) equal to 7.875 and -7.875.

Subproblem 1

x2 + -8.125 = 7.875 Simplifying x2 + -8.125 = 7.875 Reorder the terms: -8.125 + x2 = 7.875 Solving -8.125 + x2 = 7.875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8.125' to each side of the equation. -8.125 + 8.125 + x2 = 7.875 + 8.125 Combine like terms: -8.125 + 8.125 = 0.000 0.000 + x2 = 7.875 + 8.125 x2 = 7.875 + 8.125 Combine like terms: 7.875 + 8.125 = 16 x2 = 16 Simplifying x2 = 16 Take the square root of each side: x = {-4, 4}

Subproblem 2

x2 + -8.125 = -7.875 Simplifying x2 + -8.125 = -7.875 Reorder the terms: -8.125 + x2 = -7.875 Solving -8.125 + x2 = -7.875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8.125' to each side of the equation. -8.125 + 8.125 + x2 = -7.875 + 8.125 Combine like terms: -8.125 + 8.125 = 0.000 0.000 + x2 = -7.875 + 8.125 x2 = -7.875 + 8.125 Combine like terms: -7.875 + 8.125 = 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 = {-4, 4, -0.5, 0.5}

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