4x^4-37x^2+64=0

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Solution for 4x^4-37x^2+64=0 equation:


Simplifying
4x4 + -37x2 + 64 = 0

Reorder the terms:
64 + -37x2 + 4x4 = 0

Solving
64 + -37x2 + 4x4 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
16 + -9.25x2 + x4 = 0

Move the constant term to the right:

Add '-16' to each side of the equation.
16 + -9.25x2 + -16 + x4 = 0 + -16

Reorder the terms:
16 + -16 + -9.25x2 + x4 = 0 + -16

Combine like terms: 16 + -16 = 0
0 + -9.25x2 + x4 = 0 + -16
-9.25x2 + x4 = 0 + -16

Combine like terms: 0 + -16 = -16
-9.25x2 + x4 = -16

The x term is -9.25x2.  Take half its coefficient (-4.625).
Square it (21.390625) and add it to both sides.

Add '21.390625' to each side of the equation.
-9.25x2 + 21.390625 + x4 = -16 + 21.390625

Reorder the terms:
21.390625 + -9.25x2 + x4 = -16 + 21.390625

Combine like terms: -16 + 21.390625 = 5.390625
21.390625 + -9.25x2 + x4 = 5.390625

Factor a perfect square on the left side:
(x2 + -4.625)(x2 + -4.625) = 5.390625

Calculate the square root of the right side: 2.321771953

Break this problem into two subproblems by setting 
(x2 + -4.625) equal to 2.321771953 and -2.321771953.

Subproblem 1

x2 + -4.625 = 2.321771953 Simplifying x2 + -4.625 = 2.321771953 Reorder the terms: -4.625 + x2 = 2.321771953 Solving -4.625 + x2 = 2.321771953 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.625' to each side of the equation. -4.625 + 4.625 + x2 = 2.321771953 + 4.625 Combine like terms: -4.625 + 4.625 = 0.000 0.000 + x2 = 2.321771953 + 4.625 x2 = 2.321771953 + 4.625 Combine like terms: 2.321771953 + 4.625 = 6.946771953 x2 = 6.946771953 Simplifying x2 = 6.946771953 Take the square root of each side: x = {-2.63567296, 2.63567296}

Subproblem 2

x2 + -4.625 = -2.321771953 Simplifying x2 + -4.625 = -2.321771953 Reorder the terms: -4.625 + x2 = -2.321771953 Solving -4.625 + x2 = -2.321771953 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.625' to each side of the equation. -4.625 + 4.625 + x2 = -2.321771953 + 4.625 Combine like terms: -4.625 + 4.625 = 0.000 0.000 + x2 = -2.321771953 + 4.625 x2 = -2.321771953 + 4.625 Combine like terms: -2.321771953 + 4.625 = 2.303228047 x2 = 2.303228047 Simplifying x2 = 2.303228047 Take the square root of each side: x = {-1.517638971, 1.517638971}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.63567296, 2.63567296, -1.517638971, 1.517638971}

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