-7+3n(n-9)=24

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Solution for -7+3n(n-9)=24 equation:


Simplifying
-7 + 3n(n + -9) = 24

Reorder the terms:
-7 + 3n(-9 + n) = 24
-7 + (-9 * 3n + n * 3n) = 24
-7 + (-27n + 3n2) = 24

Solving
-7 + -27n + 3n2 = 24

Solving for variable 'n'.

Reorder the terms:
-7 + -24 + -27n + 3n2 = 24 + -24

Combine like terms: -7 + -24 = -31
-31 + -27n + 3n2 = 24 + -24

Combine like terms: 24 + -24 = 0
-31 + -27n + 3n2 = 0

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

Divide each side by '3'.
-10.33333333 + -9n + n2 = 0

Move the constant term to the right:

Add '10.33333333' to each side of the equation.
-10.33333333 + -9n + 10.33333333 + n2 = 0 + 10.33333333

Reorder the terms:
-10.33333333 + 10.33333333 + -9n + n2 = 0 + 10.33333333

Combine like terms: -10.33333333 + 10.33333333 = 0.00000000
0.00000000 + -9n + n2 = 0 + 10.33333333
-9n + n2 = 0 + 10.33333333

Combine like terms: 0 + 10.33333333 = 10.33333333
-9n + n2 = 10.33333333

The n term is -9n.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9n + 20.25 + n2 = 10.33333333 + 20.25

Reorder the terms:
20.25 + -9n + n2 = 10.33333333 + 20.25

Combine like terms: 10.33333333 + 20.25 = 30.58333333
20.25 + -9n + n2 = 30.58333333

Factor a perfect square on the left side:
(n + -4.5)(n + -4.5) = 30.58333333

Calculate the square root of the right side: 5.530220007

Break this problem into two subproblems by setting 
(n + -4.5) equal to 5.530220007 and -5.530220007.

Subproblem 1

n + -4.5 = 5.530220007 Simplifying n + -4.5 = 5.530220007 Reorder the terms: -4.5 + n = 5.530220007 Solving -4.5 + n = 5.530220007 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + n = 5.530220007 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = 5.530220007 + 4.5 n = 5.530220007 + 4.5 Combine like terms: 5.530220007 + 4.5 = 10.030220007 n = 10.030220007 Simplifying n = 10.030220007

Subproblem 2

n + -4.5 = -5.530220007 Simplifying n + -4.5 = -5.530220007 Reorder the terms: -4.5 + n = -5.530220007 Solving -4.5 + n = -5.530220007 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + n = -5.530220007 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = -5.530220007 + 4.5 n = -5.530220007 + 4.5 Combine like terms: -5.530220007 + 4.5 = -1.030220007 n = -1.030220007 Simplifying n = -1.030220007

Solution

The solution to the problem is based on the solutions from the subproblems. n = {10.030220007, -1.030220007}

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