x+(x+1)x=22

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Solution for x+(x+1)x=22 equation:


Simplifying
x + (x + 1) * x = 22

Reorder the terms:
x + (1 + x) * x = 22

Reorder the terms for easier multiplication:
x + x(1 + x) = 22
x + (1 * x + x * x) = 22
x + (1x + x2) = 22

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

Solving
2x + x2 = 22

Solving for variable 'x'.

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

Combine like terms: 22 + -22 = 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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