x^4-52500x^2+31250000=0

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Solution for x^4-52500x^2+31250000=0 equation:


Simplifying
x4 + -52500x2 + 31250000 = 0

Reorder the terms:
31250000 + -52500x2 + x4 = 0

Solving
31250000 + -52500x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-31250000' to each side of the equation.
31250000 + -52500x2 + -31250000 + x4 = 0 + -31250000

Reorder the terms:
31250000 + -31250000 + -52500x2 + x4 = 0 + -31250000

Combine like terms: 31250000 + -31250000 = 0
0 + -52500x2 + x4 = 0 + -31250000
-52500x2 + x4 = 0 + -31250000

Combine like terms: 0 + -31250000 = -31250000
-52500x2 + x4 = -31250000

The x term is -52500x2.  Take half its coefficient (-26250).
Square it (689062500) and add it to both sides.

Add '689062500' to each side of the equation.
-52500x2 + 689062500 + x4 = -31250000 + 689062500

Reorder the terms:
689062500 + -52500x2 + x4 = -31250000 + 689062500

Combine like terms: -31250000 + 689062500 = 657812500
689062500 + -52500x2 + x4 = 657812500

Factor a perfect square on the left side:
(x2 + -26250)(x2 + -26250) = 657812500

Calculate the square root of the right side: 25647.855660854

Break this problem into two subproblems by setting 
(x2 + -26250) equal to 25647.855660854 and -25647.855660854.

Subproblem 1

x2 + -26250 = 25647.855660854 Simplifying x2 + -26250 = 25647.855660854 Reorder the terms: -26250 + x2 = 25647.855660854 Solving -26250 + x2 = 25647.855660854 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '26250' to each side of the equation. -26250 + 26250 + x2 = 25647.855660854 + 26250 Combine like terms: -26250 + 26250 = 0 0 + x2 = 25647.855660854 + 26250 x2 = 25647.855660854 + 26250 Combine like terms: 25647.855660854 + 26250 = 51897.855660854 x2 = 51897.855660854 Simplifying x2 = 51897.855660854 Take the square root of each side: x = {-227.811008647, 227.811008647}

Subproblem 2

x2 + -26250 = -25647.855660854 Simplifying x2 + -26250 = -25647.855660854 Reorder the terms: -26250 + x2 = -25647.855660854 Solving -26250 + x2 = -25647.855660854 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '26250' to each side of the equation. -26250 + 26250 + x2 = -25647.855660854 + 26250 Combine like terms: -26250 + 26250 = 0 0 + x2 = -25647.855660854 + 26250 x2 = -25647.855660854 + 26250 Combine like terms: -25647.855660854 + 26250 = 602.144339146 x2 = 602.144339146 Simplifying x2 = 602.144339146 Take the square root of each side: x = {-24.538629529, 24.538629529}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-227.811008647, 227.811008647, -24.538629529, 24.538629529}

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