4x^4-65x^2=-64

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Solution for 4x^4-65x^2=-64 equation:


Simplifying
4x4 + -65x2 = -64

Reorder the terms:
-65x2 + 4x4 = -64

Solving
-65x2 + 4x4 = -64

Solving for variable 'x'.

Reorder the terms:
64 + -65x2 + 4x4 = -64 + 64

Combine like terms: -64 + 64 = 0
64 + -65x2 + 4x4 = 0

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

Divide each side by '4'.
16 + -16.25x2 + x4 = 0

Move the constant term to the right:

Add '-16' to each side of the equation.
16 + -16.25x2 + -16 + x4 = 0 + -16

Reorder the terms:
16 + -16 + -16.25x2 + x4 = 0 + -16

Combine like terms: 16 + -16 = 0
0 + -16.25x2 + x4 = 0 + -16
-16.25x2 + x4 = 0 + -16

Combine like terms: 0 + -16 = -16
-16.25x2 + x4 = -16

The x term is -16.25x2.  Take half its coefficient (-8.125).
Square it (66.015625) and add it to both sides.

Add '66.015625' to each side of the equation.
-16.25x2 + 66.015625 + x4 = -16 + 66.015625

Reorder the terms:
66.015625 + -16.25x2 + x4 = -16 + 66.015625

Combine like terms: -16 + 66.015625 = 50.015625
66.015625 + -16.25x2 + x4 = 50.015625

Factor a perfect square on the left side:
(x2 + -8.125)(x2 + -8.125) = 50.015625

Calculate the square root of the right side: 7.07217258

Break this problem into two subproblems by setting 
(x2 + -8.125) equal to 7.07217258 and -7.07217258.

Subproblem 1

x2 + -8.125 = 7.07217258 Simplifying x2 + -8.125 = 7.07217258 Reorder the terms: -8.125 + x2 = 7.07217258 Solving -8.125 + x2 = 7.07217258 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8.125' to each side of the equation. -8.125 + 8.125 + x2 = 7.07217258 + 8.125 Combine like terms: -8.125 + 8.125 = 0.000 0.000 + x2 = 7.07217258 + 8.125 x2 = 7.07217258 + 8.125 Combine like terms: 7.07217258 + 8.125 = 15.19717258 x2 = 15.19717258 Simplifying x2 = 15.19717258 Take the square root of each side: x = {-3.898355112, 3.898355112}

Subproblem 2

x2 + -8.125 = -7.07217258 Simplifying x2 + -8.125 = -7.07217258 Reorder the terms: -8.125 + x2 = -7.07217258 Solving -8.125 + x2 = -7.07217258 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8.125' to each side of the equation. -8.125 + 8.125 + x2 = -7.07217258 + 8.125 Combine like terms: -8.125 + 8.125 = 0.000 0.000 + x2 = -7.07217258 + 8.125 x2 = -7.07217258 + 8.125 Combine like terms: -7.07217258 + 8.125 = 1.05282742 x2 = 1.05282742 Simplifying x2 = 1.05282742 Take the square root of each side: x = {-1.026073789, 1.026073789}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.898355112, 3.898355112, -1.026073789, 1.026073789}

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