2x^2-7x-78=0

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Solution for 2x^2-7x-78=0 equation: 


Simplifying
2x2 + -7x + -78 = 0

Reorder the terms:
-78 + -7x + 2x2 = 0

Solving
-78 + -7x + 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 + -3.5x + x2 = 0

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + -3.5x + 39 + x2 = 0 + 39

Reorder the terms:
-39 + 39 + -3.5x + x2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + -3.5x + x2 = 0 + 39
-3.5x + x2 = 0 + 39

Combine like terms: 0 + 39 = 39
-3.5x + x2 = 39

The x term is -3.5x.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5x + 3.0625 + x2 = 39 + 3.0625

Reorder the terms:
3.0625 + -3.5x + x2 = 39 + 3.0625

Combine like terms: 39 + 3.0625 = 42.0625
3.0625 + -3.5x + x2 = 42.0625

Factor a perfect square on the left side:
(x + -1.75)(x + -1.75) = 42.0625

Calculate the square root of the right side: 6.485560886

Break this problem into two subproblems by setting 
(x + -1.75) equal to 6.485560886 and -6.485560886.


Subproblem 1
x + -1.75 = 6.485560886

Simplifying
x + -1.75 = 6.485560886

Reorder the terms:
-1.75 + x = 6.485560886

Solving
-1.75 + x = 6.485560886

Solving for variable 'x'.

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

Add '1.75' to each side of the equation.
-1.75 + 1.75 + x = 6.485560886 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + x = 6.485560886 + 1.75
x = 6.485560886 + 1.75

Combine like terms: 6.485560886 + 1.75 = 8.235560886
x = 8.235560886

Simplifying
x = 8.235560886


Subproblem 2
x + -1.75 = -6.485560886

Simplifying
x + -1.75 = -6.485560886

Reorder the terms:
-1.75 + x = -6.485560886

Solving
-1.75 + x = -6.485560886

Solving for variable 'x'.

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

Add '1.75' to each side of the equation.
-1.75 + 1.75 + x = -6.485560886 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + x = -6.485560886 + 1.75
x = -6.485560886 + 1.75

Combine like terms: -6.485560886 + 1.75 = -4.735560886
x = -4.735560886

Simplifying
x = -4.735560886


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8.235560886, -4.735560886}

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