x^4-52500x^2+1250000=0

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Solution for x^4-52500x^2+1250000=0 equation:


Simplifying
x4 + -52500x2 + 1250000 = 0

Reorder the terms:
1250000 + -52500x2 + x4 = 0

Solving
1250000 + -52500x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-1250000' to each side of the equation.
1250000 + -52500x2 + -1250000 + x4 = 0 + -1250000

Reorder the terms:
1250000 + -1250000 + -52500x2 + x4 = 0 + -1250000

Combine like terms: 1250000 + -1250000 = 0
0 + -52500x2 + x4 = 0 + -1250000
-52500x2 + x4 = 0 + -1250000

Combine like terms: 0 + -1250000 = -1250000
-52500x2 + x4 = -1250000

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 = -1250000 + 689062500

Reorder the terms:
689062500 + -52500x2 + x4 = -1250000 + 689062500

Combine like terms: -1250000 + 689062500 = 687812500
689062500 + -52500x2 + x4 = 687812500

Factor a perfect square on the left side:
(x2 + -26250)(x2 + -26250) = 687812500

Calculate the square root of the right side: 26226.179668415

Break this problem into two subproblems by setting 
(x2 + -26250) equal to 26226.179668415 and -26226.179668415.

Subproblem 1

x2 + -26250 = 26226.179668415 Simplifying x2 + -26250 = 26226.179668415 Reorder the terms: -26250 + x2 = 26226.179668415 Solving -26250 + x2 = 26226.179668415 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 = 26226.179668415 + 26250 Combine like terms: -26250 + 26250 = 0 0 + x2 = 26226.179668415 + 26250 x2 = 26226.179668415 + 26250 Combine like terms: 26226.179668415 + 26250 = 52476.179668415 x2 = 52476.179668415 Simplifying x2 = 52476.179668415 Take the square root of each side: x = {-229.076798625, 229.076798625}

Subproblem 2

x2 + -26250 = -26226.179668415 Simplifying x2 + -26250 = -26226.179668415 Reorder the terms: -26250 + x2 = -26226.179668415 Solving -26250 + x2 = -26226.179668415 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 = -26226.179668415 + 26250 Combine like terms: -26250 + 26250 = 0 0 + x2 = -26226.179668415 + 26250 x2 = -26226.179668415 + 26250 Combine like terms: -26226.179668415 + 26250 = 23.820331585 x2 = 23.820331585 Simplifying x2 = 23.820331585 Take the square root of each side: x = {-4.880607707, 4.880607707}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-229.076798625, 229.076798625, -4.880607707, 4.880607707}

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