x^2-4.5x-75=0

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Solution for x^2-4.5x-75=0 equation: 


Simplifying
x2 + -4.5x + -75 = 0

Reorder the terms:
-75 + -4.5x + x2 = 0

Solving
-75 + -4.5x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '75' to each side of the equation.
-75 + -4.5x + 75 + x2 = 0 + 75

Reorder the terms:
-75 + 75 + -4.5x + x2 = 0 + 75

Combine like terms: -75 + 75 = 0
0 + -4.5x + x2 = 0 + 75
-4.5x + x2 = 0 + 75

Combine like terms: 0 + 75 = 75
-4.5x + x2 = 75

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 = 75 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = 75 + 5.0625

Combine like terms: 75 + 5.0625 = 80.0625
5.0625 + -4.5x + x2 = 80.0625

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 80.0625

Calculate the square root of the right side: 8.947765084

Break this problem into two subproblems by setting 
(x + -2.25) equal to 8.947765084 and -8.947765084.


Subproblem 1
x + -2.25 = 8.947765084

Simplifying
x + -2.25 = 8.947765084

Reorder the terms:
-2.25 + x = 8.947765084

Solving
-2.25 + x = 8.947765084

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 = 8.947765084 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + x = 8.947765084 + 2.25
x = 8.947765084 + 2.25

Combine like terms: 8.947765084 + 2.25 = 11.197765084
x = 11.197765084

Simplifying
x = 11.197765084


Subproblem 2
x + -2.25 = -8.947765084

Simplifying
x + -2.25 = -8.947765084

Reorder the terms:
-2.25 + x = -8.947765084

Solving
-2.25 + x = -8.947765084

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 = -8.947765084 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + x = -8.947765084 + 2.25
x = -8.947765084 + 2.25

Combine like terms: -8.947765084 + 2.25 = -6.697765084
x = -6.697765084

Simplifying
x = -6.697765084


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {11.197765084, -6.697765084}

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