2x^2-9x-84=0

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Solution for 2x^2-9x-84=0 equation: 


Simplifying
2x2 + -9x + -84 = 0

Reorder the terms:
-84 + -9x + 2x2 = 0

Solving
-84 + -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'.
-42 + -4.5x + x2 = 0

Move the constant term to the right:

Add '42' to each side of the equation.
-42 + -4.5x + 42 + x2 = 0 + 42

Reorder the terms:
-42 + 42 + -4.5x + x2 = 0 + 42

Combine like terms: -42 + 42 = 0
0 + -4.5x + x2 = 0 + 42
-4.5x + x2 = 0 + 42

Combine like terms: 0 + 42 = 42
-4.5x + x2 = 42

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 = 42 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = 42 + 5.0625

Combine like terms: 42 + 5.0625 = 47.0625
5.0625 + -4.5x + x2 = 47.0625

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 47.0625

Calculate the square root of the right side: 6.860211367

Break this problem into two subproblems by setting 
(x + -2.25) equal to 6.860211367 and -6.860211367.


Subproblem 1
x + -2.25 = 6.860211367

Simplifying
x + -2.25 = 6.860211367

Reorder the terms:
-2.25 + x = 6.860211367

Solving
-2.25 + x = 6.860211367

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 = 6.860211367 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + x = 6.860211367 + 2.25
x = 6.860211367 + 2.25

Combine like terms: 6.860211367 + 2.25 = 9.110211367
x = 9.110211367

Simplifying
x = 9.110211367


Subproblem 2
x + -2.25 = -6.860211367

Simplifying
x + -2.25 = -6.860211367

Reorder the terms:
-2.25 + x = -6.860211367

Solving
-2.25 + x = -6.860211367

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 = -6.860211367 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + x = -6.860211367 + 2.25
x = -6.860211367 + 2.25

Combine like terms: -6.860211367 + 2.25 = -4.610211367
x = -4.610211367

Simplifying
x = -4.610211367


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {9.110211367, -4.610211367}

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