2x^2-9x-78=0

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Solution for 2x^2-9x-78=0 equation:


Simplifying
2x2 + -9x + -78 = 0

Reorder the terms:
-78 + -9x + 2x2 = 0

Solving
-78 + -9x + 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 + -4.5x + x2 = 0

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + -4.5x + 39 + x2 = 0 + 39

Reorder the terms:
-39 + 39 + -4.5x + x2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + -4.5x + x2 = 0 + 39
-4.5x + x2 = 0 + 39

Combine like terms: 0 + 39 = 39
-4.5x + x2 = 39

The x term is -4.5x.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5x + 5.0625 + x2 = 39 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = 39 + 5.0625

Combine like terms: 39 + 5.0625 = 44.0625
5.0625 + -4.5x + x2 = 44.0625

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 44.0625

Calculate the square root of the right side: 6.637959024

Break this problem into two subproblems by setting 
(x + -2.25) equal to 6.637959024 and -6.637959024.

Subproblem 1

x + -2.25 = 6.637959024 Simplifying x + -2.25 = 6.637959024 Reorder the terms: -2.25 + x = 6.637959024 Solving -2.25 + x = 6.637959024 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = 6.637959024 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = 6.637959024 + 2.25 x = 6.637959024 + 2.25 Combine like terms: 6.637959024 + 2.25 = 8.887959024 x = 8.887959024 Simplifying x = 8.887959024

Subproblem 2

x + -2.25 = -6.637959024 Simplifying x + -2.25 = -6.637959024 Reorder the terms: -2.25 + x = -6.637959024 Solving -2.25 + x = -6.637959024 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = -6.637959024 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = -6.637959024 + 2.25 x = -6.637959024 + 2.25 Combine like terms: -6.637959024 + 2.25 = -4.387959024 x = -4.387959024 Simplifying x = -4.387959024

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.887959024, -4.387959024}

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