108x^2+722x+3793=27000

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Solution for 108x^2+722x+3793=27000 equation:


Simplifying
108x2 + 722x + 3793 = 27000

Reorder the terms:
3793 + 722x + 108x2 = 27000

Solving
3793 + 722x + 108x2 = 27000

Solving for variable 'x'.

Reorder the terms:
3793 + -27000 + 722x + 108x2 = 27000 + -27000

Combine like terms: 3793 + -27000 = -23207
-23207 + 722x + 108x2 = 27000 + -27000

Combine like terms: 27000 + -27000 = 0
-23207 + 722x + 108x2 = 0

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

Divide each side by '108'.
-214.8796296 + 6.685185185x + x2 = 0

Move the constant term to the right:

Add '214.8796296' to each side of the equation.
-214.8796296 + 6.685185185x + 214.8796296 + x2 = 0 + 214.8796296

Reorder the terms:
-214.8796296 + 214.8796296 + 6.685185185x + x2 = 0 + 214.8796296

Combine like terms: -214.8796296 + 214.8796296 = 0.0000000
0.0000000 + 6.685185185x + x2 = 0 + 214.8796296
6.685185185x + x2 = 0 + 214.8796296

Combine like terms: 0 + 214.8796296 = 214.8796296
6.685185185x + x2 = 214.8796296

The x term is 6.685185185x.  Take half its coefficient (3.342592593).
Square it (11.17292524) and add it to both sides.

Add '11.17292524' to each side of the equation.
6.685185185x + 11.17292524 + x2 = 214.8796296 + 11.17292524

Reorder the terms:
11.17292524 + 6.685185185x + x2 = 214.8796296 + 11.17292524

Combine like terms: 214.8796296 + 11.17292524 = 226.05255484
11.17292524 + 6.685185185x + x2 = 226.05255484

Factor a perfect square on the left side:
(x + 3.342592593)(x + 3.342592593) = 226.05255484

Calculate the square root of the right side: 15.035044225

Break this problem into two subproblems by setting 
(x + 3.342592593) equal to 15.035044225 and -15.035044225.

Subproblem 1

x + 3.342592593 = 15.035044225 Simplifying x + 3.342592593 = 15.035044225 Reorder the terms: 3.342592593 + x = 15.035044225 Solving 3.342592593 + x = 15.035044225 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.342592593' to each side of the equation. 3.342592593 + -3.342592593 + x = 15.035044225 + -3.342592593 Combine like terms: 3.342592593 + -3.342592593 = 0.000000000 0.000000000 + x = 15.035044225 + -3.342592593 x = 15.035044225 + -3.342592593 Combine like terms: 15.035044225 + -3.342592593 = 11.692451632 x = 11.692451632 Simplifying x = 11.692451632

Subproblem 2

x + 3.342592593 = -15.035044225 Simplifying x + 3.342592593 = -15.035044225 Reorder the terms: 3.342592593 + x = -15.035044225 Solving 3.342592593 + x = -15.035044225 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.342592593' to each side of the equation. 3.342592593 + -3.342592593 + x = -15.035044225 + -3.342592593 Combine like terms: 3.342592593 + -3.342592593 = 0.000000000 0.000000000 + x = -15.035044225 + -3.342592593 x = -15.035044225 + -3.342592593 Combine like terms: -15.035044225 + -3.342592593 = -18.377636818 x = -18.377636818 Simplifying x = -18.377636818

Solution

The solution to the problem is based on the solutions from the subproblems. x = {11.692451632, -18.377636818}

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