22x^2-75=x^4

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Solution for 22x^2-75=x^4 equation:


Simplifying
22x2 + -75 = x4

Reorder the terms:
-75 + 22x2 = x4

Solving
-75 + 22x2 = x4

Solving for variable 'x'.

Combine like terms: x4 + -1x4 = 0
-75 + 22x2 + -1x4 = 0

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

Divide each side by '-1'.
75 + -22x2 + x4 = 0

Move the constant term to the right:

Add '-75' to each side of the equation.
75 + -22x2 + -75 + x4 = 0 + -75

Reorder the terms:
75 + -75 + -22x2 + x4 = 0 + -75

Combine like terms: 75 + -75 = 0
0 + -22x2 + x4 = 0 + -75
-22x2 + x4 = 0 + -75

Combine like terms: 0 + -75 = -75
-22x2 + x4 = -75

The x term is -22x2.  Take half its coefficient (-11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
-22x2 + 121 + x4 = -75 + 121

Reorder the terms:
121 + -22x2 + x4 = -75 + 121

Combine like terms: -75 + 121 = 46
121 + -22x2 + x4 = 46

Factor a perfect square on the left side:
(x2 + -11)(x2 + -11) = 46

Calculate the square root of the right side: 6.782329983

Break this problem into two subproblems by setting 
(x2 + -11) equal to 6.782329983 and -6.782329983.

Subproblem 1

x2 + -11 = 6.782329983 Simplifying x2 + -11 = 6.782329983 Reorder the terms: -11 + x2 = 6.782329983 Solving -11 + x2 = 6.782329983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + x2 = 6.782329983 + 11 Combine like terms: -11 + 11 = 0 0 + x2 = 6.782329983 + 11 x2 = 6.782329983 + 11 Combine like terms: 6.782329983 + 11 = 17.782329983 x2 = 17.782329983 Simplifying x2 = 17.782329983 Take the square root of each side: x = {-4.216910004, 4.216910004}

Subproblem 2

x2 + -11 = -6.782329983 Simplifying x2 + -11 = -6.782329983 Reorder the terms: -11 + x2 = -6.782329983 Solving -11 + x2 = -6.782329983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + x2 = -6.782329983 + 11 Combine like terms: -11 + 11 = 0 0 + x2 = -6.782329983 + 11 x2 = -6.782329983 + 11 Combine like terms: -6.782329983 + 11 = 4.217670017 x2 = 4.217670017 Simplifying x2 = 4.217670017 Take the square root of each side: x = {-2.053696671, 2.053696671}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.216910004, 4.216910004, -2.053696671, 2.053696671}

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