0=x^4-48x^2+384

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Solution for 0=x^4-48x^2+384 equation:


Simplifying
0 = x4 + -48x2 + 384

Reorder the terms:
0 = 384 + -48x2 + x4

Solving
0 = 384 + -48x2 + x4

Solving for variable 'x'.

Combine like terms: 0 + -384 = -384
-384 + 48x2 + -1x4 = 384 + -48x2 + x4 + -384 + 48x2 + -1x4

Reorder the terms:
-384 + 48x2 + -1x4 = 384 + -384 + -48x2 + 48x2 + x4 + -1x4

Combine like terms: 384 + -384 = 0
-384 + 48x2 + -1x4 = 0 + -48x2 + 48x2 + x4 + -1x4
-384 + 48x2 + -1x4 = -48x2 + 48x2 + x4 + -1x4

Combine like terms: -48x2 + 48x2 = 0
-384 + 48x2 + -1x4 = 0 + x4 + -1x4
-384 + 48x2 + -1x4 = x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-384 + 48x2 + -1x4 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
384 + -48x2 + x4 = 0

Move the constant term to the right:

Add '-384' to each side of the equation.
384 + -48x2 + -384 + x4 = 0 + -384

Reorder the terms:
384 + -384 + -48x2 + x4 = 0 + -384

Combine like terms: 384 + -384 = 0
0 + -48x2 + x4 = 0 + -384
-48x2 + x4 = 0 + -384

Combine like terms: 0 + -384 = -384
-48x2 + x4 = -384

The x term is -48x2.  Take half its coefficient (-24).
Square it (576) and add it to both sides.

Add '576' to each side of the equation.
-48x2 + 576 + x4 = -384 + 576

Reorder the terms:
576 + -48x2 + x4 = -384 + 576

Combine like terms: -384 + 576 = 192
576 + -48x2 + x4 = 192

Factor a perfect square on the left side:
(x2 + -24)(x2 + -24) = 192

Calculate the square root of the right side: 13.856406461

Break this problem into two subproblems by setting 
(x2 + -24) equal to 13.856406461 and -13.856406461.

Subproblem 1

x2 + -24 = 13.856406461 Simplifying x2 + -24 = 13.856406461 Reorder the terms: -24 + x2 = 13.856406461 Solving -24 + x2 = 13.856406461 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '24' to each side of the equation. -24 + 24 + x2 = 13.856406461 + 24 Combine like terms: -24 + 24 = 0 0 + x2 = 13.856406461 + 24 x2 = 13.856406461 + 24 Combine like terms: 13.856406461 + 24 = 37.856406461 x2 = 37.856406461 Simplifying x2 = 37.856406461 Take the square root of each side: x = {-6.152756005, 6.152756005}

Subproblem 2

x2 + -24 = -13.856406461 Simplifying x2 + -24 = -13.856406461 Reorder the terms: -24 + x2 = -13.856406461 Solving -24 + x2 = -13.856406461 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '24' to each side of the equation. -24 + 24 + x2 = -13.856406461 + 24 Combine like terms: -24 + 24 = 0 0 + x2 = -13.856406461 + 24 x2 = -13.856406461 + 24 Combine like terms: -13.856406461 + 24 = 10.143593539 x2 = 10.143593539 Simplifying x2 = 10.143593539 Take the square root of each side: x = {-3.184900868, 3.184900868}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-6.152756005, 6.152756005, -3.184900868, 3.184900868}

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