(n+6)(n-9)=90

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Solution for (n+6)(n-9)=90 equation:


Simplifying
(n + 6)(n + -9) = 90

Reorder the terms:
(6 + n)(n + -9) = 90

Reorder the terms:
(6 + n)(-9 + n) = 90

Multiply (6 + n) * (-9 + n)
(6(-9 + n) + n(-9 + n)) = 90
((-9 * 6 + n * 6) + n(-9 + n)) = 90
((-54 + 6n) + n(-9 + n)) = 90
(-54 + 6n + (-9 * n + n * n)) = 90
(-54 + 6n + (-9n + n2)) = 90

Combine like terms: 6n + -9n = -3n
(-54 + -3n + n2) = 90

Solving
-54 + -3n + n2 = 90

Solving for variable 'n'.

Reorder the terms:
-54 + -90 + -3n + n2 = 90 + -90

Combine like terms: -54 + -90 = -144
-144 + -3n + n2 = 90 + -90

Combine like terms: 90 + -90 = 0
-144 + -3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '144' to each side of the equation.
-144 + -3n + 144 + n2 = 0 + 144

Reorder the terms:
-144 + 144 + -3n + n2 = 0 + 144

Combine like terms: -144 + 144 = 0
0 + -3n + n2 = 0 + 144
-3n + n2 = 0 + 144

Combine like terms: 0 + 144 = 144
-3n + n2 = 144

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 144 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 144 + 2.25

Combine like terms: 144 + 2.25 = 146.25
2.25 + -3n + n2 = 146.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 146.25

Calculate the square root of the right side: 12.093386622

Break this problem into two subproblems by setting 
(n + -1.5) equal to 12.093386622 and -12.093386622.

Subproblem 1

n + -1.5 = 12.093386622 Simplifying n + -1.5 = 12.093386622 Reorder the terms: -1.5 + n = 12.093386622 Solving -1.5 + n = 12.093386622 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 12.093386622 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 12.093386622 + 1.5 n = 12.093386622 + 1.5 Combine like terms: 12.093386622 + 1.5 = 13.593386622 n = 13.593386622 Simplifying n = 13.593386622

Subproblem 2

n + -1.5 = -12.093386622 Simplifying n + -1.5 = -12.093386622 Reorder the terms: -1.5 + n = -12.093386622 Solving -1.5 + n = -12.093386622 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -12.093386622 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -12.093386622 + 1.5 n = -12.093386622 + 1.5 Combine like terms: -12.093386622 + 1.5 = -10.593386622 n = -10.593386622 Simplifying n = -10.593386622

Solution

The solution to the problem is based on the solutions from the subproblems. n = {13.593386622, -10.593386622}

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