x(x+1)+(x+2)=24

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Solution for x(x+1)+(x+2)=24 equation:


Simplifying
x(x + 1) + (x + 2) = 24

Reorder the terms:
x(1 + x) + (x + 2) = 24
(1 * x + x * x) + (x + 2) = 24
(1x + x2) + (x + 2) = 24

Reorder the terms:
1x + x2 + (2 + x) = 24

Remove parenthesis around (2 + x)
1x + x2 + 2 + x = 24

Reorder the terms:
2 + 1x + x + x2 = 24

Combine like terms: 1x + x = 2x
2 + 2x + x2 = 24

Solving
2 + 2x + x2 = 24

Solving for variable 'x'.

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

Combine like terms: 2 + -24 = -22
-22 + 2x + x2 = 24 + -24

Combine like terms: 24 + -24 = 0
-22 + 2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '22' to each side of the equation.
-22 + 2x + 22 + x2 = 0 + 22

Reorder the terms:
-22 + 22 + 2x + x2 = 0 + 22

Combine like terms: -22 + 22 = 0
0 + 2x + x2 = 0 + 22
2x + x2 = 0 + 22

Combine like terms: 0 + 22 = 22
2x + x2 = 22

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 = 22 + 1

Reorder the terms:
1 + 2x + x2 = 22 + 1

Combine like terms: 22 + 1 = 23
1 + 2x + x2 = 23

Factor a perfect square on the left side:
(x + 1)(x + 1) = 23

Calculate the square root of the right side: 4.795831523

Break this problem into two subproblems by setting 
(x + 1) equal to 4.795831523 and -4.795831523.

Subproblem 1

x + 1 = 4.795831523 Simplifying x + 1 = 4.795831523 Reorder the terms: 1 + x = 4.795831523 Solving 1 + x = 4.795831523 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 = 4.795831523 + -1 Combine like terms: 1 + -1 = 0 0 + x = 4.795831523 + -1 x = 4.795831523 + -1 Combine like terms: 4.795831523 + -1 = 3.795831523 x = 3.795831523 Simplifying x = 3.795831523

Subproblem 2

x + 1 = -4.795831523 Simplifying x + 1 = -4.795831523 Reorder the terms: 1 + x = -4.795831523 Solving 1 + x = -4.795831523 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 = -4.795831523 + -1 Combine like terms: 1 + -1 = 0 0 + x = -4.795831523 + -1 x = -4.795831523 + -1 Combine like terms: -4.795831523 + -1 = -5.795831523 x = -5.795831523 Simplifying x = -5.795831523

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.795831523, -5.795831523}

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