2n(n+7)=15

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Solution for 2n(n+7)=15 equation:


Simplifying
2n(n + 7) = 15

Reorder the terms:
2n(7 + n) = 15
(7 * 2n + n * 2n) = 15
(14n + 2n2) = 15

Solving
14n + 2n2 = 15

Solving for variable 'n'.

Reorder the terms:
-15 + 14n + 2n2 = 15 + -15

Combine like terms: 15 + -15 = 0
-15 + 14n + 2n2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-7.5 + 7n + n2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + 7n + 7.5 + n2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + 7n + n2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + 7n + n2 = 0 + 7.5
7n + n2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
7n + n2 = 7.5

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 = 7.5 + 12.25

Reorder the terms:
12.25 + 7n + n2 = 7.5 + 12.25

Combine like terms: 7.5 + 12.25 = 19.75
12.25 + 7n + n2 = 19.75

Factor a perfect square on the left side:
(n + 3.5)(n + 3.5) = 19.75

Calculate the square root of the right side: 4.444097209

Break this problem into two subproblems by setting 
(n + 3.5) equal to 4.444097209 and -4.444097209.

Subproblem 1

n + 3.5 = 4.444097209 Simplifying n + 3.5 = 4.444097209 Reorder the terms: 3.5 + n = 4.444097209 Solving 3.5 + n = 4.444097209 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 = 4.444097209 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 4.444097209 + -3.5 n = 4.444097209 + -3.5 Combine like terms: 4.444097209 + -3.5 = 0.944097209 n = 0.944097209 Simplifying n = 0.944097209

Subproblem 2

n + 3.5 = -4.444097209 Simplifying n + 3.5 = -4.444097209 Reorder the terms: 3.5 + n = -4.444097209 Solving 3.5 + n = -4.444097209 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 = -4.444097209 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -4.444097209 + -3.5 n = -4.444097209 + -3.5 Combine like terms: -4.444097209 + -3.5 = -7.944097209 n = -7.944097209 Simplifying n = -7.944097209

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.944097209, -7.944097209}

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