6x+12=x(x+2)-24

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Solution for 6x+12=x(x+2)-24 equation:


Simplifying
6x + 12 = x(x + 2) + -24

Reorder the terms:
12 + 6x = x(x + 2) + -24

Reorder the terms:
12 + 6x = x(2 + x) + -24
12 + 6x = (2 * x + x * x) + -24
12 + 6x = (2x + x2) + -24

Reorder the terms:
12 + 6x = -24 + 2x + x2

Solving
12 + 6x = -24 + 2x + x2

Solving for variable 'x'.

Reorder the terms:
12 + 24 + 6x + -2x + -1x2 = -24 + 2x + x2 + 24 + -2x + -1x2

Combine like terms: 12 + 24 = 36
36 + 6x + -2x + -1x2 = -24 + 2x + x2 + 24 + -2x + -1x2

Combine like terms: 6x + -2x = 4x
36 + 4x + -1x2 = -24 + 2x + x2 + 24 + -2x + -1x2

Reorder the terms:
36 + 4x + -1x2 = -24 + 24 + 2x + -2x + x2 + -1x2

Combine like terms: -24 + 24 = 0
36 + 4x + -1x2 = 0 + 2x + -2x + x2 + -1x2
36 + 4x + -1x2 = 2x + -2x + x2 + -1x2

Combine like terms: 2x + -2x = 0
36 + 4x + -1x2 = 0 + x2 + -1x2
36 + 4x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
36 + 4x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-36 + -4x + x2 = 0

Move the constant term to the right:

Add '36' to each side of the equation.
-36 + -4x + 36 + x2 = 0 + 36

Reorder the terms:
-36 + 36 + -4x + x2 = 0 + 36

Combine like terms: -36 + 36 = 0
0 + -4x + x2 = 0 + 36
-4x + x2 = 0 + 36

Combine like terms: 0 + 36 = 36
-4x + x2 = 36

The x term is -4x.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4x + 4 + x2 = 36 + 4

Reorder the terms:
4 + -4x + x2 = 36 + 4

Combine like terms: 36 + 4 = 40
4 + -4x + x2 = 40

Factor a perfect square on the left side:
(x + -2)(x + -2) = 40

Calculate the square root of the right side: 6.32455532

Break this problem into two subproblems by setting 
(x + -2) equal to 6.32455532 and -6.32455532.

Subproblem 1

x + -2 = 6.32455532 Simplifying x + -2 = 6.32455532 Reorder the terms: -2 + x = 6.32455532 Solving -2 + x = 6.32455532 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = 6.32455532 + 2 Combine like terms: -2 + 2 = 0 0 + x = 6.32455532 + 2 x = 6.32455532 + 2 Combine like terms: 6.32455532 + 2 = 8.32455532 x = 8.32455532 Simplifying x = 8.32455532

Subproblem 2

x + -2 = -6.32455532 Simplifying x + -2 = -6.32455532 Reorder the terms: -2 + x = -6.32455532 Solving -2 + x = -6.32455532 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = -6.32455532 + 2 Combine like terms: -2 + 2 = 0 0 + x = -6.32455532 + 2 x = -6.32455532 + 2 Combine like terms: -6.32455532 + 2 = -4.32455532 x = -4.32455532 Simplifying x = -4.32455532

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.32455532, -4.32455532}

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