4x^4-31x^2+56=0

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Solution for 4x^4-31x^2+56=0 equation:


Simplifying
4x4 + -31x2 + 56 = 0

Reorder the terms:
56 + -31x2 + 4x4 = 0

Solving
56 + -31x2 + 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'.
14 + -7.75x2 + x4 = 0

Move the constant term to the right:

Add '-14' to each side of the equation.
14 + -7.75x2 + -14 + x4 = 0 + -14

Reorder the terms:
14 + -14 + -7.75x2 + x4 = 0 + -14

Combine like terms: 14 + -14 = 0
0 + -7.75x2 + x4 = 0 + -14
-7.75x2 + x4 = 0 + -14

Combine like terms: 0 + -14 = -14
-7.75x2 + x4 = -14

The x term is -7.75x2.  Take half its coefficient (-3.875).
Square it (15.015625) and add it to both sides.

Add '15.015625' to each side of the equation.
-7.75x2 + 15.015625 + x4 = -14 + 15.015625

Reorder the terms:
15.015625 + -7.75x2 + x4 = -14 + 15.015625

Combine like terms: -14 + 15.015625 = 1.015625
15.015625 + -7.75x2 + x4 = 1.015625

Factor a perfect square on the left side:
(x2 + -3.875)(x2 + -3.875) = 1.015625

Calculate the square root of the right side: 1.007782219

Break this problem into two subproblems by setting 
(x2 + -3.875) equal to 1.007782219 and -1.007782219.

Subproblem 1

x2 + -3.875 = 1.007782219 Simplifying x2 + -3.875 = 1.007782219 Reorder the terms: -3.875 + x2 = 1.007782219 Solving -3.875 + x2 = 1.007782219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.875' to each side of the equation. -3.875 + 3.875 + x2 = 1.007782219 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + x2 = 1.007782219 + 3.875 x2 = 1.007782219 + 3.875 Combine like terms: 1.007782219 + 3.875 = 4.882782219 x2 = 4.882782219 Simplifying x2 = 4.882782219 Take the square root of each side: x = {-2.209701839, 2.209701839}

Subproblem 2

x2 + -3.875 = -1.007782219 Simplifying x2 + -3.875 = -1.007782219 Reorder the terms: -3.875 + x2 = -1.007782219 Solving -3.875 + x2 = -1.007782219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.875' to each side of the equation. -3.875 + 3.875 + x2 = -1.007782219 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + x2 = -1.007782219 + 3.875 x2 = -1.007782219 + 3.875 Combine like terms: -1.007782219 + 3.875 = 2.867217781 x2 = 2.867217781 Simplifying x2 = 2.867217781 Take the square root of each side: x = {-1.69328609, 1.69328609}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.209701839, 2.209701839, -1.69328609, 1.69328609}

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