x(2x+1)=19

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Solution for x(2x+1)=19 equation:


Simplifying
x(2x + 1) = 19

Reorder the terms:
x(1 + 2x) = 19
(1 * x + 2x * x) = 19
(1x + 2x2) = 19

Solving
1x + 2x2 = 19

Solving for variable 'x'.

Reorder the terms:
-19 + 1x + 2x2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + 1x + 2x2 = 0

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

Divide each side by '2'.
-9.5 + 0.5x + x2 = 0

Move the constant term to the right:

Add '9.5' to each side of the equation.
-9.5 + 0.5x + 9.5 + x2 = 0 + 9.5

Reorder the terms:
-9.5 + 9.5 + 0.5x + x2 = 0 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + 0.5x + x2 = 0 + 9.5
0.5x + x2 = 0 + 9.5

Combine like terms: 0 + 9.5 = 9.5
0.5x + x2 = 9.5

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 9.5 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 9.5 + 0.0625

Combine like terms: 9.5 + 0.0625 = 9.5625
0.0625 + 0.5x + x2 = 9.5625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 9.5625

Calculate the square root of the right side: 3.092329219

Break this problem into two subproblems by setting 
(x + 0.25) equal to 3.092329219 and -3.092329219.

Subproblem 1

x + 0.25 = 3.092329219 Simplifying x + 0.25 = 3.092329219 Reorder the terms: 0.25 + x = 3.092329219 Solving 0.25 + x = 3.092329219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 3.092329219 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 3.092329219 + -0.25 x = 3.092329219 + -0.25 Combine like terms: 3.092329219 + -0.25 = 2.842329219 x = 2.842329219 Simplifying x = 2.842329219

Subproblem 2

x + 0.25 = -3.092329219 Simplifying x + 0.25 = -3.092329219 Reorder the terms: 0.25 + x = -3.092329219 Solving 0.25 + x = -3.092329219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -3.092329219 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -3.092329219 + -0.25 x = -3.092329219 + -0.25 Combine like terms: -3.092329219 + -0.25 = -3.342329219 x = -3.342329219 Simplifying x = -3.342329219

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.842329219, -3.342329219}

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