10+3x(x+2)=31

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Solution for 10+3x(x+2)=31 equation: 


Simplifying
10 + 3x(x + 2) = 31

Reorder the terms:
10 + 3x(2 + x) = 31
10 + (2 * 3x + x * 3x) = 31
10 + (6x + 3x2) = 31

Solving
10 + 6x + 3x2 = 31

Solving for variable 'x'.

Reorder the terms:
10 + -31 + 6x + 3x2 = 31 + -31

Combine like terms: 10 + -31 = -21
-21 + 6x + 3x2 = 31 + -31

Combine like terms: 31 + -31 = 0
-21 + 6x + 3x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-7 + 2x + x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-7 + 2x + x2)' equal to zero and attempt to solve:

Simplifying
-7 + 2x + x2 = 0

Solving
-7 + 2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 2x + 7 + x2 = 0 + 7

Reorder the terms:
-7 + 7 + 2x + x2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 2x + x2 = 0 + 7
2x + x2 = 0 + 7

Combine like terms: 0 + 7 = 7
2x + x2 = 7

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 7 + 1

Reorder the terms:
1 + 2x + x2 = 7 + 1

Combine like terms: 7 + 1 = 8
1 + 2x + x2 = 8

Factor a perfect square on the left side:
(x + 1)(x + 1) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x + 1) equal to 2.828427125 and -2.828427125.


Subproblem 1
x + 1 = 2.828427125

Simplifying
x + 1 = 2.828427125

Reorder the terms:
1 + x = 2.828427125

Solving
1 + x = 2.828427125

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = 2.828427125 + -1

Combine like terms: 1 + -1 = 0
0 + x = 2.828427125 + -1
x = 2.828427125 + -1

Combine like terms: 2.828427125 + -1 = 1.828427125
x = 1.828427125

Simplifying
x = 1.828427125


Subproblem 2
x + 1 = -2.828427125

Simplifying
x + 1 = -2.828427125

Reorder the terms:
1 + x = -2.828427125

Solving
1 + x = -2.828427125

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = -2.828427125 + -1

Combine like terms: 1 + -1 = 0
0 + x = -2.828427125 + -1
x = -2.828427125 + -1

Combine like terms: -2.828427125 + -1 = -3.828427125
x = -3.828427125

Simplifying
x = -3.828427125


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.828427125, -3.828427125}
Solution
x = {1.828427125, -3.828427125}

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