x^2-41x-45=0

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Solution for x^2-41x-45=0 equation:


Simplifying
x2 + -41x + -45 = 0

Reorder the terms:
-45 + -41x + x2 = 0

Solving
-45 + -41x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '45' to each side of the equation.
-45 + -41x + 45 + x2 = 0 + 45

Reorder the terms:
-45 + 45 + -41x + x2 = 0 + 45

Combine like terms: -45 + 45 = 0
0 + -41x + x2 = 0 + 45
-41x + x2 = 0 + 45

Combine like terms: 0 + 45 = 45
-41x + x2 = 45

The x term is -41x.  Take half its coefficient (-20.5).
Square it (420.25) and add it to both sides.

Add '420.25' to each side of the equation.
-41x + 420.25 + x2 = 45 + 420.25

Reorder the terms:
420.25 + -41x + x2 = 45 + 420.25

Combine like terms: 45 + 420.25 = 465.25
420.25 + -41x + x2 = 465.25

Factor a perfect square on the left side:
(x + -20.5)(x + -20.5) = 465.25

Calculate the square root of the right side: 21.56965461

Break this problem into two subproblems by setting 
(x + -20.5) equal to 21.56965461 and -21.56965461.

Subproblem 1

x + -20.5 = 21.56965461 Simplifying x + -20.5 = 21.56965461 Reorder the terms: -20.5 + x = 21.56965461 Solving -20.5 + x = 21.56965461 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '20.5' to each side of the equation. -20.5 + 20.5 + x = 21.56965461 + 20.5 Combine like terms: -20.5 + 20.5 = 0.0 0.0 + x = 21.56965461 + 20.5 x = 21.56965461 + 20.5 Combine like terms: 21.56965461 + 20.5 = 42.06965461 x = 42.06965461 Simplifying x = 42.06965461

Subproblem 2

x + -20.5 = -21.56965461 Simplifying x + -20.5 = -21.56965461 Reorder the terms: -20.5 + x = -21.56965461 Solving -20.5 + x = -21.56965461 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '20.5' to each side of the equation. -20.5 + 20.5 + x = -21.56965461 + 20.5 Combine like terms: -20.5 + 20.5 = 0.0 0.0 + x = -21.56965461 + 20.5 x = -21.56965461 + 20.5 Combine like terms: -21.56965461 + 20.5 = -1.06965461 x = -1.06965461 Simplifying x = -1.06965461

Solution

The solution to the problem is based on the solutions from the subproblems. x = {42.06965461, -1.06965461}

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