2x^2+19x-78=0

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Solution for 2x^2+19x-78=0 equation: 


Simplifying
2x2 + 19x + -78 = 0

Reorder the terms:
-78 + 19x + 2x2 = 0

Solving
-78 + 19x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-39 + 9.5x + x2 = 0

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + 9.5x + 39 + x2 = 0 + 39

Reorder the terms:
-39 + 39 + 9.5x + x2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + 9.5x + x2 = 0 + 39
9.5x + x2 = 0 + 39

Combine like terms: 0 + 39 = 39
9.5x + x2 = 39

The x term is 9.5x.  Take half its coefficient (4.75).
Square it (22.5625) and add it to both sides.

Add '22.5625' to each side of the equation.
9.5x + 22.5625 + x2 = 39 + 22.5625

Reorder the terms:
22.5625 + 9.5x + x2 = 39 + 22.5625

Combine like terms: 39 + 22.5625 = 61.5625
22.5625 + 9.5x + x2 = 61.5625

Factor a perfect square on the left side:
(x + 4.75)(x + 4.75) = 61.5625

Calculate the square root of the right side: 7.846177413

Break this problem into two subproblems by setting 
(x + 4.75) equal to 7.846177413 and -7.846177413.


Subproblem 1
x + 4.75 = 7.846177413

Simplifying
x + 4.75 = 7.846177413

Reorder the terms:
4.75 + x = 7.846177413

Solving
4.75 + x = 7.846177413

Solving for variable 'x'.

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

Add '-4.75' to each side of the equation.
4.75 + -4.75 + x = 7.846177413 + -4.75

Combine like terms: 4.75 + -4.75 = 0.00
0.00 + x = 7.846177413 + -4.75
x = 7.846177413 + -4.75

Combine like terms: 7.846177413 + -4.75 = 3.096177413
x = 3.096177413

Simplifying
x = 3.096177413


Subproblem 2
x + 4.75 = -7.846177413

Simplifying
x + 4.75 = -7.846177413

Reorder the terms:
4.75 + x = -7.846177413

Solving
4.75 + x = -7.846177413

Solving for variable 'x'.

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

Add '-4.75' to each side of the equation.
4.75 + -4.75 + x = -7.846177413 + -4.75

Combine like terms: 4.75 + -4.75 = 0.00
0.00 + x = -7.846177413 + -4.75
x = -7.846177413 + -4.75

Combine like terms: -7.846177413 + -4.75 = -12.596177413
x = -12.596177413

Simplifying
x = -12.596177413


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.096177413, -12.596177413}

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