x^4-10500x^2+1250000=0

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Solution for x^4-10500x^2+1250000=0 equation:


Simplifying
x4 + -10500x2 + 1250000 = 0

Reorder the terms:
1250000 + -10500x2 + x4 = 0

Solving
1250000 + -10500x2 + 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 + -10500x2 + -1250000 + x4 = 0 + -1250000

Reorder the terms:
1250000 + -1250000 + -10500x2 + x4 = 0 + -1250000

Combine like terms: 1250000 + -1250000 = 0
0 + -10500x2 + x4 = 0 + -1250000
-10500x2 + x4 = 0 + -1250000

Combine like terms: 0 + -1250000 = -1250000
-10500x2 + x4 = -1250000

The x term is -10500x2.  Take half its coefficient (-5250).
Square it (27562500) and add it to both sides.

Add '27562500' to each side of the equation.
-10500x2 + 27562500 + x4 = -1250000 + 27562500

Reorder the terms:
27562500 + -10500x2 + x4 = -1250000 + 27562500

Combine like terms: -1250000 + 27562500 = 26312500
27562500 + -10500x2 + x4 = 26312500

Factor a perfect square on the left side:
(x2 + -5250)(x2 + -5250) = 26312500

Calculate the square root of the right side: 5129.571132171

Break this problem into two subproblems by setting 
(x2 + -5250) equal to 5129.571132171 and -5129.571132171.

Subproblem 1

x2 + -5250 = 5129.571132171 Simplifying x2 + -5250 = 5129.571132171 Reorder the terms: -5250 + x2 = 5129.571132171 Solving -5250 + x2 = 5129.571132171 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5250' to each side of the equation. -5250 + 5250 + x2 = 5129.571132171 + 5250 Combine like terms: -5250 + 5250 = 0 0 + x2 = 5129.571132171 + 5250 x2 = 5129.571132171 + 5250 Combine like terms: 5129.571132171 + 5250 = 10379.571132171 x2 = 10379.571132171 Simplifying x2 = 10379.571132171 Take the square root of each side: x = {-101.880180272, 101.880180272}

Subproblem 2

x2 + -5250 = -5129.571132171 Simplifying x2 + -5250 = -5129.571132171 Reorder the terms: -5250 + x2 = -5129.571132171 Solving -5250 + x2 = -5129.571132171 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5250' to each side of the equation. -5250 + 5250 + x2 = -5129.571132171 + 5250 Combine like terms: -5250 + 5250 = 0 0 + x2 = -5129.571132171 + 5250 x2 = -5129.571132171 + 5250 Combine like terms: -5129.571132171 + 5250 = 120.428867829 x2 = 120.428867829 Simplifying x2 = 120.428867829 Take the square root of each side: x = {-10.97400874, 10.97400874}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-101.880180272, 101.880180272, -10.97400874, 10.97400874}

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