x^2-5x-19=0

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Solution for x^2-5x-19=0 equation: 


Simplifying
x2 + -5x + -19 = 0

Reorder the terms:
-19 + -5x + x2 = 0

Solving
-19 + -5x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + -5x + 19 + x2 = 0 + 19

Reorder the terms:
-19 + 19 + -5x + x2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + -5x + x2 = 0 + 19
-5x + x2 = 0 + 19

Combine like terms: 0 + 19 = 19
-5x + x2 = 19

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = 19 + 6.25

Reorder the terms:
6.25 + -5x + x2 = 19 + 6.25

Combine like terms: 19 + 6.25 = 25.25
6.25 + -5x + x2 = 25.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 25.25

Calculate the square root of the right side: 5.024937811

Break this problem into two subproblems by setting 
(x + -2.5) equal to 5.024937811 and -5.024937811.


Subproblem 1
x + -2.5 = 5.024937811

Simplifying
x + -2.5 = 5.024937811

Reorder the terms:
-2.5 + x = 5.024937811

Solving
-2.5 + x = 5.024937811

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = 5.024937811 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = 5.024937811 + 2.5
x = 5.024937811 + 2.5

Combine like terms: 5.024937811 + 2.5 = 7.524937811
x = 7.524937811

Simplifying
x = 7.524937811


Subproblem 2
x + -2.5 = -5.024937811

Simplifying
x + -2.5 = -5.024937811

Reorder the terms:
-2.5 + x = -5.024937811

Solving
-2.5 + x = -5.024937811

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = -5.024937811 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = -5.024937811 + 2.5
x = -5.024937811 + 2.5

Combine like terms: -5.024937811 + 2.5 = -2.524937811
x = -2.524937811

Simplifying
x = -2.524937811


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.524937811, -2.524937811}

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