17x(2x+3)-25=366

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Solution for 17x(2x+3)-25=366 equation: 


Simplifying
17x(2x + 3) + -25 = 366

Reorder the terms:
17x(3 + 2x) + -25 = 366
(3 * 17x + 2x * 17x) + -25 = 366
(51x + 34x2) + -25 = 366

Reorder the terms:
-25 + 51x + 34x2 = 366

Solving
-25 + 51x + 34x2 = 366

Solving for variable 'x'.

Reorder the terms:
-25 + -366 + 51x + 34x2 = 366 + -366

Combine like terms: -25 + -366 = -391
-391 + 51x + 34x2 = 366 + -366

Combine like terms: 366 + -366 = 0
-391 + 51x + 34x2 = 0

Factor out the Greatest Common Factor (GCF), '17'.
17(-23 + 3x + 2x2) = 0

Ignore the factor 17.

Subproblem 1
Set the factor '(-23 + 3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-23 + 3x + 2x2 = 0

Solving
-23 + 3x + 2x2 = 0

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

Divide each side by '2'.
-11.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '11.5' to each side of the equation.
-11.5 + 1.5x + 11.5 + x2 = 0 + 11.5

Reorder the terms:
-11.5 + 11.5 + 1.5x + x2 = 0 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + 1.5x + x2 = 0 + 11.5
1.5x + x2 = 0 + 11.5

Combine like terms: 0 + 11.5 = 11.5
1.5x + x2 = 11.5

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 11.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 11.5 + 0.5625

Combine like terms: 11.5 + 0.5625 = 12.0625
0.5625 + 1.5x + x2 = 12.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 12.0625

Calculate the square root of the right side: 3.473110997

Break this problem into two subproblems by setting 
(x + 0.75) equal to 3.473110997 and -3.473110997.


Subproblem 1
x + 0.75 = 3.473110997

Simplifying
x + 0.75 = 3.473110997

Reorder the terms:
0.75 + x = 3.473110997

Solving
0.75 + x = 3.473110997

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = 3.473110997 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 3.473110997 + -0.75
x = 3.473110997 + -0.75

Combine like terms: 3.473110997 + -0.75 = 2.723110997
x = 2.723110997

Simplifying
x = 2.723110997


Subproblem 2
x + 0.75 = -3.473110997

Simplifying
x + 0.75 = -3.473110997

Reorder the terms:
0.75 + x = -3.473110997

Solving
0.75 + x = -3.473110997

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = -3.473110997 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -3.473110997 + -0.75
x = -3.473110997 + -0.75

Combine like terms: -3.473110997 + -0.75 = -4.223110997
x = -4.223110997

Simplifying
x = -4.223110997


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.723110997, -4.223110997}
Solution
x = {2.723110997, -4.223110997}

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