n^2+3n-49=0

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Solution for n^2+3n-49=0 equation:


Simplifying
n2 + 3n + -49 = 0

Reorder the terms:
-49 + 3n + n2 = 0

Solving
-49 + 3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '49' to each side of the equation.
-49 + 3n + 49 + n2 = 0 + 49

Reorder the terms:
-49 + 49 + 3n + n2 = 0 + 49

Combine like terms: -49 + 49 = 0
0 + 3n + n2 = 0 + 49
3n + n2 = 0 + 49

Combine like terms: 0 + 49 = 49
3n + n2 = 49

The n term is 3n.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3n + 2.25 + n2 = 49 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 49 + 2.25

Combine like terms: 49 + 2.25 = 51.25
2.25 + 3n + n2 = 51.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 51.25

Calculate the square root of the right side: 7.158910532

Break this problem into two subproblems by setting 
(n + 1.5) equal to 7.158910532 and -7.158910532.

Subproblem 1

n + 1.5 = 7.158910532 Simplifying n + 1.5 = 7.158910532 Reorder the terms: 1.5 + n = 7.158910532 Solving 1.5 + n = 7.158910532 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 7.158910532 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 7.158910532 + -1.5 n = 7.158910532 + -1.5 Combine like terms: 7.158910532 + -1.5 = 5.658910532 n = 5.658910532 Simplifying n = 5.658910532

Subproblem 2

n + 1.5 = -7.158910532 Simplifying n + 1.5 = -7.158910532 Reorder the terms: 1.5 + n = -7.158910532 Solving 1.5 + n = -7.158910532 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -7.158910532 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -7.158910532 + -1.5 n = -7.158910532 + -1.5 Combine like terms: -7.158910532 + -1.5 = -8.658910532 n = -8.658910532 Simplifying n = -8.658910532

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.658910532, -8.658910532}

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