32x^2-x^4=2

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


Simplifying
32x2 + -1x4 = 2

Solving
32x2 + -1x4 = 2

Solving for variable 'x'.

Reorder the terms:
-2 + 32x2 + -1x4 = 2 + -2

Combine like terms: 2 + -2 = 0
-2 + 32x2 + -1x4 = 0

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

Divide each side by '-1'.
2 + -32x2 + x4 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -32x2 + -2 + x4 = 0 + -2

Reorder the terms:
2 + -2 + -32x2 + x4 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -32x2 + x4 = 0 + -2
-32x2 + x4 = 0 + -2

Combine like terms: 0 + -2 = -2
-32x2 + x4 = -2

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 = -2 + 256

Reorder the terms:
256 + -32x2 + x4 = -2 + 256

Combine like terms: -2 + 256 = 254
256 + -32x2 + x4 = 254

Factor a perfect square on the left side:
(x2 + -16)(x2 + -16) = 254

Calculate the square root of the right side: 15.937377451

Break this problem into two subproblems by setting 
(x2 + -16) equal to 15.937377451 and -15.937377451.

Subproblem 1

x2 + -16 = 15.937377451 Simplifying x2 + -16 = 15.937377451 Reorder the terms: -16 + x2 = 15.937377451 Solving -16 + x2 = 15.937377451 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 = 15.937377451 + 16 Combine like terms: -16 + 16 = 0 0 + x2 = 15.937377451 + 16 x2 = 15.937377451 + 16 Combine like terms: 15.937377451 + 16 = 31.937377451 x2 = 31.937377451 Simplifying x2 = 31.937377451 Take the square root of each side: x = {-5.651316435, 5.651316435}

Subproblem 2

x2 + -16 = -15.937377451 Simplifying x2 + -16 = -15.937377451 Reorder the terms: -16 + x2 = -15.937377451 Solving -16 + x2 = -15.937377451 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 = -15.937377451 + 16 Combine like terms: -16 + 16 = 0 0 + x2 = -15.937377451 + 16 x2 = -15.937377451 + 16 Combine like terms: -15.937377451 + 16 = 0.062622549 x2 = 0.062622549 Simplifying x2 = 0.062622549 Take the square root of each side: x = {-0.250244978, 0.250244978}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-5.651316435, 5.651316435, -0.250244978, 0.250244978}

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