2x(x+1)+x=38

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Solution for 2x(x+1)+x=38 equation:


Simplifying
2x(x + 1) + x = 38

Reorder the terms:
2x(1 + x) + x = 38
(1 * 2x + x * 2x) + x = 38
(2x + 2x2) + x = 38

Reorder the terms:
2x + x + 2x2 = 38

Combine like terms: 2x + x = 3x
3x + 2x2 = 38

Solving
3x + 2x2 = 38

Solving for variable 'x'.

Reorder the terms:
-38 + 3x + 2x2 = 38 + -38

Combine like terms: 38 + -38 = 0
-38 + 3x + 2x2 = 0

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

Divide each side by '2'.
-19 + 1.5x + x2 = 0

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + 1.5x + 19 + x2 = 0 + 19

Reorder the terms:
-19 + 19 + 1.5x + x2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + 1.5x + x2 = 0 + 19
1.5x + x2 = 0 + 19

Combine like terms: 0 + 19 = 19
1.5x + x2 = 19

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 19 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 19 + 0.5625

Combine like terms: 19 + 0.5625 = 19.5625
0.5625 + 1.5x + x2 = 19.5625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 19.5625

Calculate the square root of the right side: 4.422951503

Break this problem into two subproblems by setting 
(x + 0.75) equal to 4.422951503 and -4.422951503.

Subproblem 1

x + 0.75 = 4.422951503 Simplifying x + 0.75 = 4.422951503 Reorder the terms: 0.75 + x = 4.422951503 Solving 0.75 + x = 4.422951503 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 4.422951503 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 4.422951503 + -0.75 x = 4.422951503 + -0.75 Combine like terms: 4.422951503 + -0.75 = 3.672951503 x = 3.672951503 Simplifying x = 3.672951503

Subproblem 2

x + 0.75 = -4.422951503 Simplifying x + 0.75 = -4.422951503 Reorder the terms: 0.75 + x = -4.422951503 Solving 0.75 + x = -4.422951503 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -4.422951503 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -4.422951503 + -0.75 x = -4.422951503 + -0.75 Combine like terms: -4.422951503 + -0.75 = -5.172951503 x = -5.172951503 Simplifying x = -5.172951503

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.672951503, -5.172951503}

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