5x^4-51x^2+16=0

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Solution for 5x^4-51x^2+16=0 equation:


Simplifying
5x4 + -51x2 + 16 = 0

Reorder the terms:
16 + -51x2 + 5x4 = 0

Solving
16 + -51x2 + 5x4 = 0

Solving for variable 'x'.

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

Divide each side by '5'.
3.2 + -10.2x2 + x4 = 0

Move the constant term to the right:

Add '-3.2' to each side of the equation.
3.2 + -10.2x2 + -3.2 + x4 = 0 + -3.2

Reorder the terms:
3.2 + -3.2 + -10.2x2 + x4 = 0 + -3.2

Combine like terms: 3.2 + -3.2 = 0.0
0.0 + -10.2x2 + x4 = 0 + -3.2
-10.2x2 + x4 = 0 + -3.2

Combine like terms: 0 + -3.2 = -3.2
-10.2x2 + x4 = -3.2

The x term is -10.2x2.  Take half its coefficient (-5.1).
Square it (26.01) and add it to both sides.

Add '26.01' to each side of the equation.
-10.2x2 + 26.01 + x4 = -3.2 + 26.01

Reorder the terms:
26.01 + -10.2x2 + x4 = -3.2 + 26.01

Combine like terms: -3.2 + 26.01 = 22.81
26.01 + -10.2x2 + x4 = 22.81

Factor a perfect square on the left side:
(x2 + -5.1)(x2 + -5.1) = 22.81

Calculate the square root of the right side: 4.775981575

Break this problem into two subproblems by setting 
(x2 + -5.1) equal to 4.775981575 and -4.775981575.

Subproblem 1

x2 + -5.1 = 4.775981575 Simplifying x2 + -5.1 = 4.775981575 Reorder the terms: -5.1 + x2 = 4.775981575 Solving -5.1 + x2 = 4.775981575 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.1' to each side of the equation. -5.1 + 5.1 + x2 = 4.775981575 + 5.1 Combine like terms: -5.1 + 5.1 = 0.0 0.0 + x2 = 4.775981575 + 5.1 x2 = 4.775981575 + 5.1 Combine like terms: 4.775981575 + 5.1 = 9.875981575 x2 = 9.875981575 Simplifying x2 = 9.875981575 Take the square root of each side: x = {-3.142607448, 3.142607448}

Subproblem 2

x2 + -5.1 = -4.775981575 Simplifying x2 + -5.1 = -4.775981575 Reorder the terms: -5.1 + x2 = -4.775981575 Solving -5.1 + x2 = -4.775981575 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.1' to each side of the equation. -5.1 + 5.1 + x2 = -4.775981575 + 5.1 Combine like terms: -5.1 + 5.1 = 0.0 0.0 + x2 = -4.775981575 + 5.1 x2 = -4.775981575 + 5.1 Combine like terms: -4.775981575 + 5.1 = 0.324018425 x2 = 0.324018425 Simplifying x2 = 0.324018425 Take the square root of each side: x = {-0.569226163, 0.569226163}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.142607448, 3.142607448, -0.569226163, 0.569226163}

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