n(2n+3)=75

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Solution for n(2n+3)=75 equation:


Simplifying
n(2n + 3) = 75

Reorder the terms:
n(3 + 2n) = 75
(3 * n + 2n * n) = 75
(3n + 2n2) = 75

Solving
3n + 2n2 = 75

Solving for variable 'n'.

Reorder the terms:
-75 + 3n + 2n2 = 75 + -75

Combine like terms: 75 + -75 = 0
-75 + 3n + 2n2 = 0

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

Divide each side by '2'.
-37.5 + 1.5n + n2 = 0

Move the constant term to the right:

Add '37.5' to each side of the equation.
-37.5 + 1.5n + 37.5 + n2 = 0 + 37.5

Reorder the terms:
-37.5 + 37.5 + 1.5n + n2 = 0 + 37.5

Combine like terms: -37.5 + 37.5 = 0.0
0.0 + 1.5n + n2 = 0 + 37.5
1.5n + n2 = 0 + 37.5

Combine like terms: 0 + 37.5 = 37.5
1.5n + n2 = 37.5

The n term is 1.5n.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5n + 0.5625 + n2 = 37.5 + 0.5625

Reorder the terms:
0.5625 + 1.5n + n2 = 37.5 + 0.5625

Combine like terms: 37.5 + 0.5625 = 38.0625
0.5625 + 1.5n + n2 = 38.0625

Factor a perfect square on the left side:
(n + 0.75)(n + 0.75) = 38.0625

Calculate the square root of the right side: 6.16948134

Break this problem into two subproblems by setting 
(n + 0.75) equal to 6.16948134 and -6.16948134.

Subproblem 1

n + 0.75 = 6.16948134 Simplifying n + 0.75 = 6.16948134 Reorder the terms: 0.75 + n = 6.16948134 Solving 0.75 + n = 6.16948134 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = 6.16948134 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = 6.16948134 + -0.75 n = 6.16948134 + -0.75 Combine like terms: 6.16948134 + -0.75 = 5.41948134 n = 5.41948134 Simplifying n = 5.41948134

Subproblem 2

n + 0.75 = -6.16948134 Simplifying n + 0.75 = -6.16948134 Reorder the terms: 0.75 + n = -6.16948134 Solving 0.75 + n = -6.16948134 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = -6.16948134 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = -6.16948134 + -0.75 n = -6.16948134 + -0.75 Combine like terms: -6.16948134 + -0.75 = -6.91948134 n = -6.91948134 Simplifying n = -6.91948134

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.41948134, -6.91948134}

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