n^2+19n-40=0

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Solution for n^2+19n-40=0 equation:


Simplifying
n2 + 19n + -40 = 0

Reorder the terms:
-40 + 19n + n2 = 0

Solving
-40 + 19n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + 19n + 40 + n2 = 0 + 40

Reorder the terms:
-40 + 40 + 19n + n2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + 19n + n2 = 0 + 40
19n + n2 = 0 + 40

Combine like terms: 0 + 40 = 40
19n + n2 = 40

The n term is 19n.  Take half its coefficient (9.5).
Square it (90.25) and add it to both sides.

Add '90.25' to each side of the equation.
19n + 90.25 + n2 = 40 + 90.25

Reorder the terms:
90.25 + 19n + n2 = 40 + 90.25

Combine like terms: 40 + 90.25 = 130.25
90.25 + 19n + n2 = 130.25

Factor a perfect square on the left side:
(n + 9.5)(n + 9.5) = 130.25

Calculate the square root of the right side: 11.412712211

Break this problem into two subproblems by setting 
(n + 9.5) equal to 11.412712211 and -11.412712211.

Subproblem 1

n + 9.5 = 11.412712211 Simplifying n + 9.5 = 11.412712211 Reorder the terms: 9.5 + n = 11.412712211 Solving 9.5 + n = 11.412712211 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + n = 11.412712211 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + n = 11.412712211 + -9.5 n = 11.412712211 + -9.5 Combine like terms: 11.412712211 + -9.5 = 1.912712211 n = 1.912712211 Simplifying n = 1.912712211

Subproblem 2

n + 9.5 = -11.412712211 Simplifying n + 9.5 = -11.412712211 Reorder the terms: 9.5 + n = -11.412712211 Solving 9.5 + n = -11.412712211 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + n = -11.412712211 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + n = -11.412712211 + -9.5 n = -11.412712211 + -9.5 Combine like terms: -11.412712211 + -9.5 = -20.912712211 n = -20.912712211 Simplifying n = -20.912712211

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.912712211, -20.912712211}

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