7x(7+x)=112

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Solution for 7x(7+x)=112 equation: 


Simplifying
7x(7 + x) = 112
(7 * 7x + x * 7x) = 112
(49x + 7x2) = 112

Solving
49x + 7x2 = 112

Solving for variable 'x'.

Reorder the terms:
-112 + 49x + 7x2 = 112 + -112

Combine like terms: 112 + -112 = 0
-112 + 49x + 7x2 = 0

Factor out the Greatest Common Factor (GCF), '7'.
7(-16 + 7x + x2) = 0

Ignore the factor 7.

Subproblem 1
Set the factor '(-16 + 7x + x2)' equal to zero and attempt to solve:

Simplifying
-16 + 7x + x2 = 0

Solving
-16 + 7x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + 7x + 16 + x2 = 0 + 16

Reorder the terms:
-16 + 16 + 7x + x2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + 7x + x2 = 0 + 16
7x + x2 = 0 + 16

Combine like terms: 0 + 16 = 16
7x + x2 = 16

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = 16 + 12.25

Reorder the terms:
12.25 + 7x + x2 = 16 + 12.25

Combine like terms: 16 + 12.25 = 28.25
12.25 + 7x + x2 = 28.25

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 28.25

Calculate the square root of the right side: 5.315072906

Break this problem into two subproblems by setting 
(x + 3.5) equal to 5.315072906 and -5.315072906.


Subproblem 1
x + 3.5 = 5.315072906

Simplifying
x + 3.5 = 5.315072906

Reorder the terms:
3.5 + x = 5.315072906

Solving
3.5 + x = 5.315072906

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = 5.315072906 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = 5.315072906 + -3.5
x = 5.315072906 + -3.5

Combine like terms: 5.315072906 + -3.5 = 1.815072906
x = 1.815072906

Simplifying
x = 1.815072906


Subproblem 2
x + 3.5 = -5.315072906

Simplifying
x + 3.5 = -5.315072906

Reorder the terms:
3.5 + x = -5.315072906

Solving
3.5 + x = -5.315072906

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = -5.315072906 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = -5.315072906 + -3.5
x = -5.315072906 + -3.5

Combine like terms: -5.315072906 + -3.5 = -8.815072906
x = -8.815072906

Simplifying
x = -8.815072906


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.815072906, -8.815072906}
Solution
x = {1.815072906, -8.815072906}

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