2x^2+x=19

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Solution for 2x^2+x=19 equation:


Simplifying
2x2 + x = 19

Reorder the terms:
x + 2x2 = 19

Solving
x + 2x2 = 19

Solving for variable 'x'.

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

Combine like terms: 19 + -19 = 0
-19 + x + 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 x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.5x + 0.25 + x2 = 9.5 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 9.5 + 0.25

Combine like terms: 9.5 + 0.25 = 9.75
0.25 + 0.5x + x2 = 9.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 9.75

Calculate the square root of the right side: 3.122498999

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.122498999 and -3.122498999.

Subproblem 1

x + 0.5 = 3.122498999 Simplifying x + 0.5 = 3.122498999 Reorder the terms: 0.5 + x = 3.122498999 Solving 0.5 + x = 3.122498999 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 3.122498999 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 3.122498999 + -0.5 x = 3.122498999 + -0.5 Combine like terms: 3.122498999 + -0.5 = 2.622498999 x = 2.622498999 Simplifying x = 2.622498999

Subproblem 2

x + 0.5 = -3.122498999 Simplifying x + 0.5 = -3.122498999 Reorder the terms: 0.5 + x = -3.122498999 Solving 0.5 + x = -3.122498999 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -3.122498999 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -3.122498999 + -0.5 x = -3.122498999 + -0.5 Combine like terms: -3.122498999 + -0.5 = -3.622498999 x = -3.622498999 Simplifying x = -3.622498999

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.622498999, -3.622498999}

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