n^2+7n=240

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Solution for n^2+7n=240 equation:


Simplifying
n2 + 7n = 240

Reorder the terms:
7n + n2 = 240

Solving
7n + n2 = 240

Solving for variable 'n'.

Reorder the terms:
-240 + 7n + n2 = 240 + -240

Combine like terms: 240 + -240 = 0
-240 + 7n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '240' to each side of the equation.
-240 + 7n + 240 + n2 = 0 + 240

Reorder the terms:
-240 + 240 + 7n + n2 = 0 + 240

Combine like terms: -240 + 240 = 0
0 + 7n + n2 = 0 + 240
7n + n2 = 0 + 240

Combine like terms: 0 + 240 = 240
7n + n2 = 240

The n term is 7n.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7n + 12.25 + n2 = 240 + 12.25

Reorder the terms:
12.25 + 7n + n2 = 240 + 12.25

Combine like terms: 240 + 12.25 = 252.25
12.25 + 7n + n2 = 252.25

Factor a perfect square on the left side:
(n + 3.5)(n + 3.5) = 252.25

Calculate the square root of the right side: 15.882380174

Break this problem into two subproblems by setting 
(n + 3.5) equal to 15.882380174 and -15.882380174.

Subproblem 1

n + 3.5 = 15.882380174 Simplifying n + 3.5 = 15.882380174 Reorder the terms: 3.5 + n = 15.882380174 Solving 3.5 + n = 15.882380174 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = 15.882380174 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 15.882380174 + -3.5 n = 15.882380174 + -3.5 Combine like terms: 15.882380174 + -3.5 = 12.382380174 n = 12.382380174 Simplifying n = 12.382380174

Subproblem 2

n + 3.5 = -15.882380174 Simplifying n + 3.5 = -15.882380174 Reorder the terms: 3.5 + n = -15.882380174 Solving 3.5 + n = -15.882380174 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = -15.882380174 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -15.882380174 + -3.5 n = -15.882380174 + -3.5 Combine like terms: -15.882380174 + -3.5 = -19.382380174 n = -19.382380174 Simplifying n = -19.382380174

Solution

The solution to the problem is based on the solutions from the subproblems. n = {12.382380174, -19.382380174}

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