9x^2-85x+109=0

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Solution for 9x^2-85x+109=0 equation:


Simplifying
9x2 + -85x + 109 = 0

Reorder the terms:
109 + -85x + 9x2 = 0

Solving
109 + -85x + 9x2 = 0

Solving for variable 'x'.

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

Divide each side by '9'.
12.11111111 + -9.444444444x + x2 = 0

Move the constant term to the right:

Add '-12.11111111' to each side of the equation.
12.11111111 + -9.444444444x + -12.11111111 + x2 = 0 + -12.11111111

Reorder the terms:
12.11111111 + -12.11111111 + -9.444444444x + x2 = 0 + -12.11111111

Combine like terms: 12.11111111 + -12.11111111 = 0.00000000
0.00000000 + -9.444444444x + x2 = 0 + -12.11111111
-9.444444444x + x2 = 0 + -12.11111111

Combine like terms: 0 + -12.11111111 = -12.11111111
-9.444444444x + x2 = -12.11111111

The x term is -9.444444444x.  Take half its coefficient (-4.722222222).
Square it (22.29938271) and add it to both sides.

Add '22.29938271' to each side of the equation.
-9.444444444x + 22.29938271 + x2 = -12.11111111 + 22.29938271

Reorder the terms:
22.29938271 + -9.444444444x + x2 = -12.11111111 + 22.29938271

Combine like terms: -12.11111111 + 22.29938271 = 10.1882716
22.29938271 + -9.444444444x + x2 = 10.1882716

Factor a perfect square on the left side:
(x + -4.722222222)(x + -4.722222222) = 10.1882716

Calculate the square root of the right side: 3.191907204

Break this problem into two subproblems by setting 
(x + -4.722222222) equal to 3.191907204 and -3.191907204.

Subproblem 1

x + -4.722222222 = 3.191907204 Simplifying x + -4.722222222 = 3.191907204 Reorder the terms: -4.722222222 + x = 3.191907204 Solving -4.722222222 + x = 3.191907204 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.722222222' to each side of the equation. -4.722222222 + 4.722222222 + x = 3.191907204 + 4.722222222 Combine like terms: -4.722222222 + 4.722222222 = 0.000000000 0.000000000 + x = 3.191907204 + 4.722222222 x = 3.191907204 + 4.722222222 Combine like terms: 3.191907204 + 4.722222222 = 7.914129426 x = 7.914129426 Simplifying x = 7.914129426

Subproblem 2

x + -4.722222222 = -3.191907204 Simplifying x + -4.722222222 = -3.191907204 Reorder the terms: -4.722222222 + x = -3.191907204 Solving -4.722222222 + x = -3.191907204 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.722222222' to each side of the equation. -4.722222222 + 4.722222222 + x = -3.191907204 + 4.722222222 Combine like terms: -4.722222222 + 4.722222222 = 0.000000000 0.000000000 + x = -3.191907204 + 4.722222222 x = -3.191907204 + 4.722222222 Combine like terms: -3.191907204 + 4.722222222 = 1.530315018 x = 1.530315018 Simplifying x = 1.530315018

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.914129426, 1.530315018}

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