8.5-7.09x+1.09x^2=0

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Solution for 8.5-7.09x+1.09x^2=0 equation:


Simplifying
8.5 + -7.09x + 1.09x2 = 0

Solving
8.5 + -7.09x + 1.09x2 = 0

Solving for variable 'x'.

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

Divide each side by '1.09'.
7.798165138 + -6.504587156x + x2 = 0

Move the constant term to the right:

Add '-7.798165138' to each side of the equation.
7.798165138 + -6.504587156x + -7.798165138 + x2 = 0 + -7.798165138

Reorder the terms:
7.798165138 + -7.798165138 + -6.504587156x + x2 = 0 + -7.798165138

Combine like terms: 7.798165138 + -7.798165138 = 0.000000000
0.000000000 + -6.504587156x + x2 = 0 + -7.798165138
-6.504587156x + x2 = 0 + -7.798165138

Combine like terms: 0 + -7.798165138 = -7.798165138
-6.504587156x + x2 = -7.798165138

The x term is -6.504587156x.  Take half its coefficient (-3.252293578).
Square it (10.57741352) and add it to both sides.

Add '10.57741352' to each side of the equation.
-6.504587156x + 10.57741352 + x2 = -7.798165138 + 10.57741352

Reorder the terms:
10.57741352 + -6.504587156x + x2 = -7.798165138 + 10.57741352

Combine like terms: -7.798165138 + 10.57741352 = 2.779248382
10.57741352 + -6.504587156x + x2 = 2.779248382

Factor a perfect square on the left side:
(x + -3.252293578)(x + -3.252293578) = 2.779248382

Calculate the square root of the right side: 1.66710779

Break this problem into two subproblems by setting 
(x + -3.252293578) equal to 1.66710779 and -1.66710779.


Subproblem 1
x + -3.252293578 = 1.66710779

Simplifying
x + -3.252293578 = 1.66710779

Reorder the terms:
-3.252293578 + x = 1.66710779

Solving
-3.252293578 + x = 1.66710779

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '3.252293578' to each side of the equation.
-3.252293578 + 3.252293578 + x = 1.66710779 + 3.252293578

Combine like terms: -3.252293578 + 3.252293578 = 0.000000000
0.000000000 + x = 1.66710779 + 3.252293578
x = 1.66710779 + 3.252293578

Combine like terms: 1.66710779 + 3.252293578 = 4.919401368
x = 4.919401368

Simplifying
x = 4.919401368


Subproblem 2
x + -3.252293578 = -1.66710779

Simplifying
x + -3.252293578 = -1.66710779

Reorder the terms:
-3.252293578 + x = -1.66710779

Solving
-3.252293578 + x = -1.66710779

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '3.252293578' to each side of the equation.
-3.252293578 + 3.252293578 + x = -1.66710779 + 3.252293578

Combine like terms: -3.252293578 + 3.252293578 = 0.000000000
0.000000000 + x = -1.66710779 + 3.252293578
x = -1.66710779 + 3.252293578

Combine like terms: -1.66710779 + 3.252293578 = 1.585185788
x = 1.585185788

Simplifying
x = 1.585185788


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.919401368, 1.585185788}

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