(n+5)(n-6)=21

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Solution for (n+5)(n-6)=21 equation:


Simplifying
(n + 5)(n + -6) = 21

Reorder the terms:
(5 + n)(n + -6) = 21

Reorder the terms:
(5 + n)(-6 + n) = 21

Multiply (5 + n) * (-6 + n)
(5(-6 + n) + n(-6 + n)) = 21
((-6 * 5 + n * 5) + n(-6 + n)) = 21
((-30 + 5n) + n(-6 + n)) = 21
(-30 + 5n + (-6 * n + n * n)) = 21
(-30 + 5n + (-6n + n2)) = 21

Combine like terms: 5n + -6n = -1n
(-30 + -1n + n2) = 21

Solving
-30 + -1n + n2 = 21

Solving for variable 'n'.

Reorder the terms:
-30 + -21 + -1n + n2 = 21 + -21

Combine like terms: -30 + -21 = -51
-51 + -1n + n2 = 21 + -21

Combine like terms: 21 + -21 = 0
-51 + -1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '51' to each side of the equation.
-51 + -1n + 51 + n2 = 0 + 51

Reorder the terms:
-51 + 51 + -1n + n2 = 0 + 51

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

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

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.
-1n + 0.25 + n2 = 51 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 51 + 0.25

Combine like terms: 51 + 0.25 = 51.25
0.25 + -1n + n2 = 51.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 51.25

Calculate the square root of the right side: 7.158910532

Break this problem into two subproblems by setting 
(n + -0.5) equal to 7.158910532 and -7.158910532.

Subproblem 1

n + -0.5 = 7.158910532 Simplifying n + -0.5 = 7.158910532 Reorder the terms: -0.5 + n = 7.158910532 Solving -0.5 + n = 7.158910532 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 = 7.158910532 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 7.158910532 + 0.5 n = 7.158910532 + 0.5 Combine like terms: 7.158910532 + 0.5 = 7.658910532 n = 7.658910532 Simplifying n = 7.658910532

Subproblem 2

n + -0.5 = -7.158910532 Simplifying n + -0.5 = -7.158910532 Reorder the terms: -0.5 + n = -7.158910532 Solving -0.5 + n = -7.158910532 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 = -7.158910532 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -7.158910532 + 0.5 n = -7.158910532 + 0.5 Combine like terms: -7.158910532 + 0.5 = -6.658910532 n = -6.658910532 Simplifying n = -6.658910532

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.658910532, -6.658910532}

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