(5+x)(7+x)=105

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Solution for (5+x)(7+x)=105 equation: 


Simplifying
(5 + x)(7 + x) = 105

Multiply (5 + x) * (7 + x)
(5(7 + x) + x(7 + x)) = 105
((7 * 5 + x * 5) + x(7 + x)) = 105
((35 + 5x) + x(7 + x)) = 105
(35 + 5x + (7 * x + x * x)) = 105
(35 + 5x + (7x + x2)) = 105

Combine like terms: 5x + 7x = 12x
(35 + 12x + x2) = 105

Solving
35 + 12x + x2 = 105

Solving for variable 'x'.

Reorder the terms:
35 + -105 + 12x + x2 = 105 + -105

Combine like terms: 35 + -105 = -70
-70 + 12x + x2 = 105 + -105

Combine like terms: 105 + -105 = 0
-70 + 12x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '70' to each side of the equation.
-70 + 12x + 70 + x2 = 0 + 70

Reorder the terms:
-70 + 70 + 12x + x2 = 0 + 70

Combine like terms: -70 + 70 = 0
0 + 12x + x2 = 0 + 70
12x + x2 = 0 + 70

Combine like terms: 0 + 70 = 70
12x + x2 = 70

The x term is 12x.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12x + 36 + x2 = 70 + 36

Reorder the terms:
36 + 12x + x2 = 70 + 36

Combine like terms: 70 + 36 = 106
36 + 12x + x2 = 106

Factor a perfect square on the left side:
(x + 6)(x + 6) = 106

Calculate the square root of the right side: 10.295630141

Break this problem into two subproblems by setting 
(x + 6) equal to 10.295630141 and -10.295630141.


Subproblem 1
x + 6 = 10.295630141

Simplifying
x + 6 = 10.295630141

Reorder the terms:
6 + x = 10.295630141

Solving
6 + x = 10.295630141

Solving for variable 'x'.

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

Add '-6' to each side of the equation.
6 + -6 + x = 10.295630141 + -6

Combine like terms: 6 + -6 = 0
0 + x = 10.295630141 + -6
x = 10.295630141 + -6

Combine like terms: 10.295630141 + -6 = 4.295630141
x = 4.295630141

Simplifying
x = 4.295630141


Subproblem 2
x + 6 = -10.295630141

Simplifying
x + 6 = -10.295630141

Reorder the terms:
6 + x = -10.295630141

Solving
6 + x = -10.295630141

Solving for variable 'x'.

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

Add '-6' to each side of the equation.
6 + -6 + x = -10.295630141 + -6

Combine like terms: 6 + -6 = 0
0 + x = -10.295630141 + -6
x = -10.295630141 + -6

Combine like terms: -10.295630141 + -6 = -16.295630141
x = -16.295630141

Simplifying
x = -16.295630141


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.295630141, -16.295630141}

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