x^2-45x-40=0

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Solution for x^2-45x-40=0 equation:


Simplifying
x2 + -45x + -40 = 0

Reorder the terms:
-40 + -45x + x2 = 0

Solving
-40 + -45x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + -45x + 40 + x2 = 0 + 40

Reorder the terms:
-40 + 40 + -45x + x2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + -45x + x2 = 0 + 40
-45x + x2 = 0 + 40

Combine like terms: 0 + 40 = 40
-45x + x2 = 40

The x term is -45x.  Take half its coefficient (-22.5).
Square it (506.25) and add it to both sides.

Add '506.25' to each side of the equation.
-45x + 506.25 + x2 = 40 + 506.25

Reorder the terms:
506.25 + -45x + x2 = 40 + 506.25

Combine like terms: 40 + 506.25 = 546.25
506.25 + -45x + x2 = 546.25

Factor a perfect square on the left side:
(x + -22.5)(x + -22.5) = 546.25

Calculate the square root of the right side: 23.371991785

Break this problem into two subproblems by setting 
(x + -22.5) equal to 23.371991785 and -23.371991785.


Subproblem 1
x + -22.5 = 23.371991785

Simplifying
x + -22.5 = 23.371991785

Reorder the terms:
-22.5 + x = 23.371991785

Solving
-22.5 + x = 23.371991785

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '22.5' to each side of the equation.
-22.5 + 22.5 + x = 23.371991785 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + x = 23.371991785 + 22.5
x = 23.371991785 + 22.5

Combine like terms: 23.371991785 + 22.5 = 45.871991785
x = 45.871991785

Simplifying
x = 45.871991785


Subproblem 2
x + -22.5 = -23.371991785

Simplifying
x + -22.5 = -23.371991785

Reorder the terms:
-22.5 + x = -23.371991785

Solving
-22.5 + x = -23.371991785

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '22.5' to each side of the equation.
-22.5 + 22.5 + x = -23.371991785 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + x = -23.371991785 + 22.5
x = -23.371991785 + 22.5

Combine like terms: -23.371991785 + 22.5 = -0.871991785
x = -0.871991785

Simplifying
x = -0.871991785


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {45.871991785, -0.871991785}

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