x(x+1)+(x+2)=51

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Solution for x(x+1)+(x+2)=51 equation:


Simplifying
x(x + 1) + (x + 2) = 51

Reorder the terms:
x(1 + x) + (x + 2) = 51
(1 * x + x * x) + (x + 2) = 51
(1x + x2) + (x + 2) = 51

Reorder the terms:
1x + x2 + (2 + x) = 51

Remove parenthesis around (2 + x)
1x + x2 + 2 + x = 51

Reorder the terms:
2 + 1x + x + x2 = 51

Combine like terms: 1x + x = 2x
2 + 2x + x2 = 51

Solving
2 + 2x + x2 = 51

Solving for variable 'x'.

Reorder the terms:
2 + -51 + 2x + x2 = 51 + -51

Combine like terms: 2 + -51 = -49
-49 + 2x + x2 = 51 + -51

Combine like terms: 51 + -51 = 0
-49 + 2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '49' to each side of the equation.
-49 + 2x + 49 + x2 = 0 + 49

Reorder the terms:
-49 + 49 + 2x + x2 = 0 + 49

Combine like terms: -49 + 49 = 0
0 + 2x + x2 = 0 + 49
2x + x2 = 0 + 49

Combine like terms: 0 + 49 = 49
2x + x2 = 49

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 = 49 + 1

Reorder the terms:
1 + 2x + x2 = 49 + 1

Combine like terms: 49 + 1 = 50
1 + 2x + x2 = 50

Factor a perfect square on the left side:
(x + 1)(x + 1) = 50

Calculate the square root of the right side: 7.071067812

Break this problem into two subproblems by setting 
(x + 1) equal to 7.071067812 and -7.071067812.

Subproblem 1

x + 1 = 7.071067812 Simplifying x + 1 = 7.071067812 Reorder the terms: 1 + x = 7.071067812 Solving 1 + x = 7.071067812 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 = 7.071067812 + -1 Combine like terms: 1 + -1 = 0 0 + x = 7.071067812 + -1 x = 7.071067812 + -1 Combine like terms: 7.071067812 + -1 = 6.071067812 x = 6.071067812 Simplifying x = 6.071067812

Subproblem 2

x + 1 = -7.071067812 Simplifying x + 1 = -7.071067812 Reorder the terms: 1 + x = -7.071067812 Solving 1 + x = -7.071067812 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 = -7.071067812 + -1 Combine like terms: 1 + -1 = 0 0 + x = -7.071067812 + -1 x = -7.071067812 + -1 Combine like terms: -7.071067812 + -1 = -8.071067812 x = -8.071067812 Simplifying x = -8.071067812

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.071067812, -8.071067812}

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