5x^4-24x^2+1=0

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Solution for 5x^4-24x^2+1=0 equation:


Simplifying
5x4 + -24x2 + 1 = 0

Reorder the terms:
1 + -24x2 + 5x4 = 0

Solving
1 + -24x2 + 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'.
0.2 + -4.8x2 + x4 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + -4.8x2 + -0.2 + x4 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + -4.8x2 + x4 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + -4.8x2 + x4 = 0 + -0.2
-4.8x2 + x4 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
-4.8x2 + x4 = -0.2

The x term is -4.8x2.  Take half its coefficient (-2.4).
Square it (5.76) and add it to both sides.

Add '5.76' to each side of the equation.
-4.8x2 + 5.76 + x4 = -0.2 + 5.76

Reorder the terms:
5.76 + -4.8x2 + x4 = -0.2 + 5.76

Combine like terms: -0.2 + 5.76 = 5.56
5.76 + -4.8x2 + x4 = 5.56

Factor a perfect square on the left side:
(x2 + -2.4)(x2 + -2.4) = 5.56

Calculate the square root of the right side: 2.357965225

Break this problem into two subproblems by setting 
(x2 + -2.4) equal to 2.357965225 and -2.357965225.

Subproblem 1

x2 + -2.4 = 2.357965225 Simplifying x2 + -2.4 = 2.357965225 Reorder the terms: -2.4 + x2 = 2.357965225 Solving -2.4 + x2 = 2.357965225 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.4' to each side of the equation. -2.4 + 2.4 + x2 = 2.357965225 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + x2 = 2.357965225 + 2.4 x2 = 2.357965225 + 2.4 Combine like terms: 2.357965225 + 2.4 = 4.757965225 x2 = 4.757965225 Simplifying x2 = 4.757965225 Take the square root of each side: x = {-2.181276054, 2.181276054}

Subproblem 2

x2 + -2.4 = -2.357965225 Simplifying x2 + -2.4 = -2.357965225 Reorder the terms: -2.4 + x2 = -2.357965225 Solving -2.4 + x2 = -2.357965225 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.4' to each side of the equation. -2.4 + 2.4 + x2 = -2.357965225 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + x2 = -2.357965225 + 2.4 x2 = -2.357965225 + 2.4 Combine like terms: -2.357965225 + 2.4 = 0.042034775 x2 = 0.042034775 Simplifying x2 = 0.042034775 Take the square root of each side: x = {-0.20502384, 0.20502384}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.181276054, 2.181276054, -0.20502384, 0.20502384}

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