x^4-32x^2=-225

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Solution for x^4-32x^2=-225 equation:


Simplifying
x4 + -32x2 = -225

Reorder the terms:
-32x2 + x4 = -225

Solving
-32x2 + x4 = -225

Solving for variable 'x'.

Reorder the terms:
225 + -32x2 + x4 = -225 + 225

Combine like terms: -225 + 225 = 0
225 + -32x2 + x4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-225' to each side of the equation.
225 + -32x2 + -225 + x4 = 0 + -225

Reorder the terms:
225 + -225 + -32x2 + x4 = 0 + -225

Combine like terms: 225 + -225 = 0
0 + -32x2 + x4 = 0 + -225
-32x2 + x4 = 0 + -225

Combine like terms: 0 + -225 = -225
-32x2 + x4 = -225

The x term is -32x2.  Take half its coefficient (-16).
Square it (256) and add it to both sides.

Add '256' to each side of the equation.
-32x2 + 256 + x4 = -225 + 256

Reorder the terms:
256 + -32x2 + x4 = -225 + 256

Combine like terms: -225 + 256 = 31
256 + -32x2 + x4 = 31

Factor a perfect square on the left side:
(x2 + -16)(x2 + -16) = 31

Calculate the square root of the right side: 5.567764363

Break this problem into two subproblems by setting 
(x2 + -16) equal to 5.567764363 and -5.567764363.

Subproblem 1

x2 + -16 = 5.567764363 Simplifying x2 + -16 = 5.567764363 Reorder the terms: -16 + x2 = 5.567764363 Solving -16 + x2 = 5.567764363 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + x2 = 5.567764363 + 16 Combine like terms: -16 + 16 = 0 0 + x2 = 5.567764363 + 16 x2 = 5.567764363 + 16 Combine like terms: 5.567764363 + 16 = 21.567764363 x2 = 21.567764363 Simplifying x2 = 21.567764363 Take the square root of each side: x = {-4.644110718, 4.644110718}

Subproblem 2

x2 + -16 = -5.567764363 Simplifying x2 + -16 = -5.567764363 Reorder the terms: -16 + x2 = -5.567764363 Solving -16 + x2 = -5.567764363 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + x2 = -5.567764363 + 16 Combine like terms: -16 + 16 = 0 0 + x2 = -5.567764363 + 16 x2 = -5.567764363 + 16 Combine like terms: -5.567764363 + 16 = 10.432235637 x2 = 10.432235637 Simplifying x2 = 10.432235637 Take the square root of each side: x = {-3.229897156, 3.229897156}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.644110718, 4.644110718, -3.229897156, 3.229897156}

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