x^4-21x^2+64=0

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Solution for x^4-21x^2+64=0 equation:


Simplifying
x4 + -21x2 + 64 = 0

Reorder the terms:
64 + -21x2 + x4 = 0

Solving
64 + -21x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-64' to each side of the equation.
64 + -21x2 + -64 + x4 = 0 + -64

Reorder the terms:
64 + -64 + -21x2 + x4 = 0 + -64

Combine like terms: 64 + -64 = 0
0 + -21x2 + x4 = 0 + -64
-21x2 + x4 = 0 + -64

Combine like terms: 0 + -64 = -64
-21x2 + x4 = -64

The x term is -21x2.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21x2 + 110.25 + x4 = -64 + 110.25

Reorder the terms:
110.25 + -21x2 + x4 = -64 + 110.25

Combine like terms: -64 + 110.25 = 46.25
110.25 + -21x2 + x4 = 46.25

Factor a perfect square on the left side:
(x2 + -10.5)(x2 + -10.5) = 46.25

Calculate the square root of the right side: 6.800735254

Break this problem into two subproblems by setting 
(x2 + -10.5) equal to 6.800735254 and -6.800735254.

Subproblem 1

x2 + -10.5 = 6.800735254 Simplifying x2 + -10.5 = 6.800735254 Reorder the terms: -10.5 + x2 = 6.800735254 Solving -10.5 + x2 = 6.800735254 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + x2 = 6.800735254 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x2 = 6.800735254 + 10.5 x2 = 6.800735254 + 10.5 Combine like terms: 6.800735254 + 10.5 = 17.300735254 x2 = 17.300735254 Simplifying x2 = 17.300735254 Take the square root of each side: x = {-4.159415254, 4.159415254}

Subproblem 2

x2 + -10.5 = -6.800735254 Simplifying x2 + -10.5 = -6.800735254 Reorder the terms: -10.5 + x2 = -6.800735254 Solving -10.5 + x2 = -6.800735254 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + x2 = -6.800735254 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x2 = -6.800735254 + 10.5 x2 = -6.800735254 + 10.5 Combine like terms: -6.800735254 + 10.5 = 3.699264746 x2 = 3.699264746 Simplifying x2 = 3.699264746 Take the square root of each side: x = {-1.923347276, 1.923347276}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.159415254, 4.159415254, -1.923347276, 1.923347276}

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