x^2+2.5x-22.5=0

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Solution for x^2+2.5x-22.5=0 equation: 


Simplifying
x2 + 2.5x + -22.5 = 0

Reorder the terms:
-22.5 + 2.5x + x2 = 0

Solving
-22.5 + 2.5x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + 2.5x + 22.5 + x2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + 2.5x + x2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + 2.5x + x2 = 0 + 22.5
2.5x + x2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
2.5x + x2 = 22.5

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 = 22.5 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = 22.5 + 1.5625

Combine like terms: 22.5 + 1.5625 = 24.0625
1.5625 + 2.5x + x2 = 24.0625

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 24.0625

Calculate the square root of the right side: 4.905354218

Break this problem into two subproblems by setting 
(x + 1.25) equal to 4.905354218 and -4.905354218.


Subproblem 1
x + 1.25 = 4.905354218

Simplifying
x + 1.25 = 4.905354218

Reorder the terms:
1.25 + x = 4.905354218

Solving
1.25 + x = 4.905354218

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 = 4.905354218 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x = 4.905354218 + -1.25
x = 4.905354218 + -1.25

Combine like terms: 4.905354218 + -1.25 = 3.655354218
x = 3.655354218

Simplifying
x = 3.655354218


Subproblem 2
x + 1.25 = -4.905354218

Simplifying
x + 1.25 = -4.905354218

Reorder the terms:
1.25 + x = -4.905354218

Solving
1.25 + x = -4.905354218

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 = -4.905354218 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x = -4.905354218 + -1.25
x = -4.905354218 + -1.25

Combine like terms: -4.905354218 + -1.25 = -6.155354218
x = -6.155354218

Simplifying
x = -6.155354218


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.655354218, -6.155354218}

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