58x-2x^2=21x+109

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Solution for 58x-2x^2=21x+109 equation: 


Simplifying
58x + -2x2 = 21x + 109

Reorder the terms:
58x + -2x2 = 109 + 21x

Solving
58x + -2x2 = 109 + 21x

Solving for variable 'x'.

Reorder the terms:
-109 + 58x + -21x + -2x2 = 109 + 21x + -109 + -21x

Combine like terms: 58x + -21x = 37x
-109 + 37x + -2x2 = 109 + 21x + -109 + -21x

Reorder the terms:
-109 + 37x + -2x2 = 109 + -109 + 21x + -21x

Combine like terms: 109 + -109 = 0
-109 + 37x + -2x2 = 0 + 21x + -21x
-109 + 37x + -2x2 = 21x + -21x

Combine like terms: 21x + -21x = 0
-109 + 37x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
54.5 + -18.5x + x2 = 0

Move the constant term to the right:

Add '-54.5' to each side of the equation.
54.5 + -18.5x + -54.5 + x2 = 0 + -54.5

Reorder the terms:
54.5 + -54.5 + -18.5x + x2 = 0 + -54.5

Combine like terms: 54.5 + -54.5 = 0.0
0.0 + -18.5x + x2 = 0 + -54.5
-18.5x + x2 = 0 + -54.5

Combine like terms: 0 + -54.5 = -54.5
-18.5x + x2 = -54.5

The x term is -18.5x.  Take half its coefficient (-9.25).
Square it (85.5625) and add it to both sides.

Add '85.5625' to each side of the equation.
-18.5x + 85.5625 + x2 = -54.5 + 85.5625

Reorder the terms:
85.5625 + -18.5x + x2 = -54.5 + 85.5625

Combine like terms: -54.5 + 85.5625 = 31.0625
85.5625 + -18.5x + x2 = 31.0625

Factor a perfect square on the left side:
(x + -9.25)(x + -9.25) = 31.0625

Calculate the square root of the right side: 5.573374202

Break this problem into two subproblems by setting 
(x + -9.25) equal to 5.573374202 and -5.573374202.


Subproblem 1
x + -9.25 = 5.573374202

Simplifying
x + -9.25 = 5.573374202

Reorder the terms:
-9.25 + x = 5.573374202

Solving
-9.25 + x = 5.573374202

Solving for variable 'x'.

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

Add '9.25' to each side of the equation.
-9.25 + 9.25 + x = 5.573374202 + 9.25

Combine like terms: -9.25 + 9.25 = 0.00
0.00 + x = 5.573374202 + 9.25
x = 5.573374202 + 9.25

Combine like terms: 5.573374202 + 9.25 = 14.823374202
x = 14.823374202

Simplifying
x = 14.823374202


Subproblem 2
x + -9.25 = -5.573374202

Simplifying
x + -9.25 = -5.573374202

Reorder the terms:
-9.25 + x = -5.573374202

Solving
-9.25 + x = -5.573374202

Solving for variable 'x'.

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

Add '9.25' to each side of the equation.
-9.25 + 9.25 + x = -5.573374202 + 9.25

Combine like terms: -9.25 + 9.25 = 0.00
0.00 + x = -5.573374202 + 9.25
x = -5.573374202 + 9.25

Combine like terms: -5.573374202 + 9.25 = 3.676625798
x = 3.676625798

Simplifying
x = 3.676625798


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {14.823374202, 3.676625798}

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