(27+2x)(10+2x)=78

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Solution for (27+2x)(10+2x)=78 equation: 


Simplifying
(27 + 2x)(10 + 2x) = 78

Multiply (27 + 2x) * (10 + 2x)
(27(10 + 2x) + 2x * (10 + 2x)) = 78
((10 * 27 + 2x * 27) + 2x * (10 + 2x)) = 78
((270 + 54x) + 2x * (10 + 2x)) = 78
(270 + 54x + (10 * 2x + 2x * 2x)) = 78
(270 + 54x + (20x + 4x2)) = 78

Combine like terms: 54x + 20x = 74x
(270 + 74x + 4x2) = 78

Solving
270 + 74x + 4x2 = 78

Solving for variable 'x'.

Reorder the terms:
270 + -78 + 74x + 4x2 = 78 + -78

Combine like terms: 270 + -78 = 192
192 + 74x + 4x2 = 78 + -78

Combine like terms: 78 + -78 = 0
192 + 74x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(96 + 37x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(96 + 37x + 2x2)' equal to zero and attempt to solve:

Simplifying
96 + 37x + 2x2 = 0

Solving
96 + 37x + 2x2 = 0

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

Divide each side by '2'.
48 + 18.5x + x2 = 0

Move the constant term to the right:

Add '-48' to each side of the equation.
48 + 18.5x + -48 + x2 = 0 + -48

Reorder the terms:
48 + -48 + 18.5x + x2 = 0 + -48

Combine like terms: 48 + -48 = 0
0 + 18.5x + x2 = 0 + -48
18.5x + x2 = 0 + -48

Combine like terms: 0 + -48 = -48
18.5x + x2 = -48

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 = -48 + 85.5625

Reorder the terms:
85.5625 + 18.5x + x2 = -48 + 85.5625

Combine like terms: -48 + 85.5625 = 37.5625
85.5625 + 18.5x + x2 = 37.5625

Factor a perfect square on the left side:
(x + 9.25)(x + 9.25) = 37.5625

Calculate the square root of the right side: 6.128825336

Break this problem into two subproblems by setting 
(x + 9.25) equal to 6.128825336 and -6.128825336.


Subproblem 1
x + 9.25 = 6.128825336

Simplifying
x + 9.25 = 6.128825336

Reorder the terms:
9.25 + x = 6.128825336

Solving
9.25 + x = 6.128825336

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 = 6.128825336 + -9.25

Combine like terms: 9.25 + -9.25 = 0.00
0.00 + x = 6.128825336 + -9.25
x = 6.128825336 + -9.25

Combine like terms: 6.128825336 + -9.25 = -3.121174664
x = -3.121174664

Simplifying
x = -3.121174664


Subproblem 2
x + 9.25 = -6.128825336

Simplifying
x + 9.25 = -6.128825336

Reorder the terms:
9.25 + x = -6.128825336

Solving
9.25 + x = -6.128825336

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 = -6.128825336 + -9.25

Combine like terms: 9.25 + -9.25 = 0.00
0.00 + x = -6.128825336 + -9.25
x = -6.128825336 + -9.25

Combine like terms: -6.128825336 + -9.25 = -15.378825336
x = -15.378825336

Simplifying
x = -15.378825336


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-3.121174664, -15.378825336}
Solution
x = {-3.121174664, -15.378825336}

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