2n(n-3)=5(n+3)

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Solution for 2n(n-3)=5(n+3) equation:


Simplifying
2n(n + -3) = 5(n + 3)

Reorder the terms:
2n(-3 + n) = 5(n + 3)
(-3 * 2n + n * 2n) = 5(n + 3)
(-6n + 2n2) = 5(n + 3)

Reorder the terms:
-6n + 2n2 = 5(3 + n)
-6n + 2n2 = (3 * 5 + n * 5)
-6n + 2n2 = (15 + 5n)

Solving
-6n + 2n2 = 15 + 5n

Solving for variable 'n'.

Reorder the terms:
-15 + -6n + -5n + 2n2 = 15 + 5n + -15 + -5n

Combine like terms: -6n + -5n = -11n
-15 + -11n + 2n2 = 15 + 5n + -15 + -5n

Reorder the terms:
-15 + -11n + 2n2 = 15 + -15 + 5n + -5n

Combine like terms: 15 + -15 = 0
-15 + -11n + 2n2 = 0 + 5n + -5n
-15 + -11n + 2n2 = 5n + -5n

Combine like terms: 5n + -5n = 0
-15 + -11n + 2n2 = 0

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

Divide each side by '2'.
-7.5 + -5.5n + n2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + -5.5n + 7.5 + n2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + -5.5n + n2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + -5.5n + n2 = 0 + 7.5
-5.5n + n2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
-5.5n + n2 = 7.5

The n term is -5.5n.  Take half its coefficient (-2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
-5.5n + 7.5625 + n2 = 7.5 + 7.5625

Reorder the terms:
7.5625 + -5.5n + n2 = 7.5 + 7.5625

Combine like terms: 7.5 + 7.5625 = 15.0625
7.5625 + -5.5n + n2 = 15.0625

Factor a perfect square on the left side:
(n + -2.75)(n + -2.75) = 15.0625

Calculate the square root of the right side: 3.881043674

Break this problem into two subproblems by setting 
(n + -2.75) equal to 3.881043674 and -3.881043674.

Subproblem 1

n + -2.75 = 3.881043674 Simplifying n + -2.75 = 3.881043674 Reorder the terms: -2.75 + n = 3.881043674 Solving -2.75 + n = 3.881043674 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + n = 3.881043674 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + n = 3.881043674 + 2.75 n = 3.881043674 + 2.75 Combine like terms: 3.881043674 + 2.75 = 6.631043674 n = 6.631043674 Simplifying n = 6.631043674

Subproblem 2

n + -2.75 = -3.881043674 Simplifying n + -2.75 = -3.881043674 Reorder the terms: -2.75 + n = -3.881043674 Solving -2.75 + n = -3.881043674 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + n = -3.881043674 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + n = -3.881043674 + 2.75 n = -3.881043674 + 2.75 Combine like terms: -3.881043674 + 2.75 = -1.131043674 n = -1.131043674 Simplifying n = -1.131043674

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.631043674, -1.131043674}

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