(n+5)(n+5-9)=51

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Solution for (n+5)(n+5-9)=51 equation:


Simplifying
(n + 5)(n + 5 + -9) = 51

Reorder the terms:
(5 + n)(n + 5 + -9) = 51

Reorder the terms:
(5 + n)(5 + -9 + n) = 51

Combine like terms: 5 + -9 = -4
(5 + n)(-4 + n) = 51

Multiply (5 + n) * (-4 + n)
(5(-4 + n) + n(-4 + n)) = 51
((-4 * 5 + n * 5) + n(-4 + n)) = 51
((-20 + 5n) + n(-4 + n)) = 51
(-20 + 5n + (-4 * n + n * n)) = 51
(-20 + 5n + (-4n + n2)) = 51

Combine like terms: 5n + -4n = 1n
(-20 + 1n + n2) = 51

Solving
-20 + 1n + n2 = 51

Solving for variable 'n'.

Reorder the terms:
-20 + -51 + 1n + n2 = 51 + -51

Combine like terms: -20 + -51 = -71
-71 + 1n + n2 = 51 + -51

Combine like terms: 51 + -51 = 0
-71 + 1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '71' to each side of the equation.
-71 + 1n + 71 + n2 = 0 + 71

Reorder the terms:
-71 + 71 + 1n + n2 = 0 + 71

Combine like terms: -71 + 71 = 0
0 + 1n + n2 = 0 + 71
1n + n2 = 0 + 71

Combine like terms: 0 + 71 = 71
1n + n2 = 71

The n term is 1n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
n + 0.25 + n2 = 71 + 0.25

Reorder the terms:
0.25 + n + n2 = 71 + 0.25

Combine like terms: 71 + 0.25 = 71.25
0.25 + n + n2 = 71.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 71.25

Calculate the square root of the right side: 8.440971508

Break this problem into two subproblems by setting 
(n + 0.5) equal to 8.440971508 and -8.440971508.

Subproblem 1

n + 0.5 = 8.440971508 Simplifying n + 0.5 = 8.440971508 Reorder the terms: 0.5 + n = 8.440971508 Solving 0.5 + n = 8.440971508 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 8.440971508 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 8.440971508 + -0.5 n = 8.440971508 + -0.5 Combine like terms: 8.440971508 + -0.5 = 7.940971508 n = 7.940971508 Simplifying n = 7.940971508

Subproblem 2

n + 0.5 = -8.440971508 Simplifying n + 0.5 = -8.440971508 Reorder the terms: 0.5 + n = -8.440971508 Solving 0.5 + n = -8.440971508 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -8.440971508 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -8.440971508 + -0.5 n = -8.440971508 + -0.5 Combine like terms: -8.440971508 + -0.5 = -8.940971508 n = -8.940971508 Simplifying n = -8.940971508

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.940971508, -8.940971508}

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