2x^2-5x-78=0

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Solution for 2x^2-5x-78=0 equation:


Simplifying
2x2 + -5x + -78 = 0

Reorder the terms:
-78 + -5x + 2x2 = 0

Solving
-78 + -5x + 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 + -2.5x + x2 = 0

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + -2.5x + 39 + x2 = 0 + 39

Reorder the terms:
-39 + 39 + -2.5x + x2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + -2.5x + x2 = 0 + 39
-2.5x + x2 = 0 + 39

Combine like terms: 0 + 39 = 39
-2.5x + x2 = 39

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = 39 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = 39 + 1.5625

Combine like terms: 39 + 1.5625 = 40.5625
1.5625 + -2.5x + x2 = 40.5625

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 40.5625

Calculate the square root of the right side: 6.368869601

Break this problem into two subproblems by setting 
(x + -1.25) equal to 6.368869601 and -6.368869601.

Subproblem 1

x + -1.25 = 6.368869601 Simplifying x + -1.25 = 6.368869601 Reorder the terms: -1.25 + x = 6.368869601 Solving -1.25 + x = 6.368869601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 6.368869601 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 6.368869601 + 1.25 x = 6.368869601 + 1.25 Combine like terms: 6.368869601 + 1.25 = 7.618869601 x = 7.618869601 Simplifying x = 7.618869601

Subproblem 2

x + -1.25 = -6.368869601 Simplifying x + -1.25 = -6.368869601 Reorder the terms: -1.25 + x = -6.368869601 Solving -1.25 + x = -6.368869601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -6.368869601 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -6.368869601 + 1.25 x = -6.368869601 + 1.25 Combine like terms: -6.368869601 + 1.25 = -5.118869601 x = -5.118869601 Simplifying x = -5.118869601

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.618869601, -5.118869601}

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