25x^4-291x^2+675=0

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Solution for 25x^4-291x^2+675=0 equation:


Simplifying
25x4 + -291x2 + 675 = 0

Reorder the terms:
675 + -291x2 + 25x4 = 0

Solving
675 + -291x2 + 25x4 = 0

Solving for variable 'x'.

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

Divide each side by '25'.
27 + -11.64x2 + x4 = 0

Move the constant term to the right:

Add '-27' to each side of the equation.
27 + -11.64x2 + -27 + x4 = 0 + -27

Reorder the terms:
27 + -27 + -11.64x2 + x4 = 0 + -27

Combine like terms: 27 + -27 = 0
0 + -11.64x2 + x4 = 0 + -27
-11.64x2 + x4 = 0 + -27

Combine like terms: 0 + -27 = -27
-11.64x2 + x4 = -27

The x term is -11.64x2.  Take half its coefficient (-5.82).
Square it (33.8724) and add it to both sides.

Add '33.8724' to each side of the equation.
-11.64x2 + 33.8724 + x4 = -27 + 33.8724

Reorder the terms:
33.8724 + -11.64x2 + x4 = -27 + 33.8724

Combine like terms: -27 + 33.8724 = 6.8724
33.8724 + -11.64x2 + x4 = 6.8724

Factor a perfect square on the left side:
(x2 + -5.82)(x2 + -5.82) = 6.8724

Calculate the square root of the right side: 2.621526273

Break this problem into two subproblems by setting 
(x2 + -5.82) equal to 2.621526273 and -2.621526273.

Subproblem 1

x2 + -5.82 = 2.621526273 Simplifying x2 + -5.82 = 2.621526273 Reorder the terms: -5.82 + x2 = 2.621526273 Solving -5.82 + x2 = 2.621526273 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.82' to each side of the equation. -5.82 + 5.82 + x2 = 2.621526273 + 5.82 Combine like terms: -5.82 + 5.82 = 0.00 0.00 + x2 = 2.621526273 + 5.82 x2 = 2.621526273 + 5.82 Combine like terms: 2.621526273 + 5.82 = 8.441526273 x2 = 8.441526273 Simplifying x2 = 8.441526273 Take the square root of each side: x = {-2.90543048, 2.90543048}

Subproblem 2

x2 + -5.82 = -2.621526273 Simplifying x2 + -5.82 = -2.621526273 Reorder the terms: -5.82 + x2 = -2.621526273 Solving -5.82 + x2 = -2.621526273 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.82' to each side of the equation. -5.82 + 5.82 + x2 = -2.621526273 + 5.82 Combine like terms: -5.82 + 5.82 = 0.00 0.00 + x2 = -2.621526273 + 5.82 x2 = -2.621526273 + 5.82 Combine like terms: -2.621526273 + 5.82 = 3.198473727 x2 = 3.198473727 Simplifying x2 = 3.198473727 Take the square root of each side: x = {-1.788427725, 1.788427725}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.90543048, 2.90543048, -1.788427725, 1.788427725}

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