x^2+41x+42=0

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Solution for x^2+41x+42=0 equation:


Simplifying
x2 + 41x + 42 = 0

Reorder the terms:
42 + 41x + x2 = 0

Solving
42 + 41x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-42' to each side of the equation.
42 + 41x + -42 + x2 = 0 + -42

Reorder the terms:
42 + -42 + 41x + x2 = 0 + -42

Combine like terms: 42 + -42 = 0
0 + 41x + x2 = 0 + -42
41x + x2 = 0 + -42

Combine like terms: 0 + -42 = -42
41x + x2 = -42

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 = -42 + 420.25

Reorder the terms:
420.25 + 41x + x2 = -42 + 420.25

Combine like terms: -42 + 420.25 = 378.25
420.25 + 41x + x2 = 378.25

Factor a perfect square on the left side:
(x + 20.5)(x + 20.5) = 378.25

Calculate the square root of the right side: 19.448650339

Break this problem into two subproblems by setting 
(x + 20.5) equal to 19.448650339 and -19.448650339.

Subproblem 1

x + 20.5 = 19.448650339 Simplifying x + 20.5 = 19.448650339 Reorder the terms: 20.5 + x = 19.448650339 Solving 20.5 + x = 19.448650339 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 = 19.448650339 + -20.5 Combine like terms: 20.5 + -20.5 = 0.0 0.0 + x = 19.448650339 + -20.5 x = 19.448650339 + -20.5 Combine like terms: 19.448650339 + -20.5 = -1.051349661 x = -1.051349661 Simplifying x = -1.051349661

Subproblem 2

x + 20.5 = -19.448650339 Simplifying x + 20.5 = -19.448650339 Reorder the terms: 20.5 + x = -19.448650339 Solving 20.5 + x = -19.448650339 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 = -19.448650339 + -20.5 Combine like terms: 20.5 + -20.5 = 0.0 0.0 + x = -19.448650339 + -20.5 x = -19.448650339 + -20.5 Combine like terms: -19.448650339 + -20.5 = -39.948650339 x = -39.948650339 Simplifying x = -39.948650339

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.051349661, -39.948650339}

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