n^2+5n+6=70

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Solution for n^2+5n+6=70 equation:


Simplifying
n2 + 5n + 6 = 70

Reorder the terms:
6 + 5n + n2 = 70

Solving
6 + 5n + n2 = 70

Solving for variable 'n'.

Reorder the terms:
6 + -70 + 5n + n2 = 70 + -70

Combine like terms: 6 + -70 = -64
-64 + 5n + n2 = 70 + -70

Combine like terms: 70 + -70 = 0
-64 + 5n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '64' to each side of the equation.
-64 + 5n + 64 + n2 = 0 + 64

Reorder the terms:
-64 + 64 + 5n + n2 = 0 + 64

Combine like terms: -64 + 64 = 0
0 + 5n + n2 = 0 + 64
5n + n2 = 0 + 64

Combine like terms: 0 + 64 = 64
5n + n2 = 64

The n term is 5n.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5n + 6.25 + n2 = 64 + 6.25

Reorder the terms:
6.25 + 5n + n2 = 64 + 6.25

Combine like terms: 64 + 6.25 = 70.25
6.25 + 5n + n2 = 70.25

Factor a perfect square on the left side:
(n + 2.5)(n + 2.5) = 70.25

Calculate the square root of the right side: 8.381527307

Break this problem into two subproblems by setting 
(n + 2.5) equal to 8.381527307 and -8.381527307.

Subproblem 1

n + 2.5 = 8.381527307 Simplifying n + 2.5 = 8.381527307 Reorder the terms: 2.5 + n = 8.381527307 Solving 2.5 + n = 8.381527307 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 8.381527307 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 8.381527307 + -2.5 n = 8.381527307 + -2.5 Combine like terms: 8.381527307 + -2.5 = 5.881527307 n = 5.881527307 Simplifying n = 5.881527307

Subproblem 2

n + 2.5 = -8.381527307 Simplifying n + 2.5 = -8.381527307 Reorder the terms: 2.5 + n = -8.381527307 Solving 2.5 + n = -8.381527307 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -8.381527307 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -8.381527307 + -2.5 n = -8.381527307 + -2.5 Combine like terms: -8.381527307 + -2.5 = -10.881527307 n = -10.881527307 Simplifying n = -10.881527307

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.881527307, -10.881527307}

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