x^2+4.5x-53=0

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Solution for x^2+4.5x-53=0 equation: 


Simplifying
x2 + 4.5x + -53 = 0

Reorder the terms:
-53 + 4.5x + x2 = 0

Solving
-53 + 4.5x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '53' to each side of the equation.
-53 + 4.5x + 53 + x2 = 0 + 53

Reorder the terms:
-53 + 53 + 4.5x + x2 = 0 + 53

Combine like terms: -53 + 53 = 0
0 + 4.5x + x2 = 0 + 53
4.5x + x2 = 0 + 53

Combine like terms: 0 + 53 = 53
4.5x + x2 = 53

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 = 53 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = 53 + 5.0625

Combine like terms: 53 + 5.0625 = 58.0625
5.0625 + 4.5x + x2 = 58.0625

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 58.0625

Calculate the square root of the right side: 7.619875327

Break this problem into two subproblems by setting 
(x + 2.25) equal to 7.619875327 and -7.619875327.


Subproblem 1
x + 2.25 = 7.619875327

Simplifying
x + 2.25 = 7.619875327

Reorder the terms:
2.25 + x = 7.619875327

Solving
2.25 + x = 7.619875327

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 = 7.619875327 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + x = 7.619875327 + -2.25
x = 7.619875327 + -2.25

Combine like terms: 7.619875327 + -2.25 = 5.369875327
x = 5.369875327

Simplifying
x = 5.369875327


Subproblem 2
x + 2.25 = -7.619875327

Simplifying
x + 2.25 = -7.619875327

Reorder the terms:
2.25 + x = -7.619875327

Solving
2.25 + x = -7.619875327

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 = -7.619875327 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + x = -7.619875327 + -2.25
x = -7.619875327 + -2.25

Combine like terms: -7.619875327 + -2.25 = -9.869875327
x = -9.869875327

Simplifying
x = -9.869875327


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.369875327, -9.869875327}

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