2n^2+11n=75

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Solution for 2n^2+11n=75 equation:


Simplifying
2n2 + 11n = 75

Reorder the terms:
11n + 2n2 = 75

Solving
11n + 2n2 = 75

Solving for variable 'n'.

Reorder the terms:
-75 + 11n + 2n2 = 75 + -75

Combine like terms: 75 + -75 = 0
-75 + 11n + 2n2 = 0

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

Divide each side by '2'.
-37.5 + 5.5n + n2 = 0

Move the constant term to the right:

Add '37.5' to each side of the equation.
-37.5 + 5.5n + 37.5 + n2 = 0 + 37.5

Reorder the terms:
-37.5 + 37.5 + 5.5n + n2 = 0 + 37.5

Combine like terms: -37.5 + 37.5 = 0.0
0.0 + 5.5n + n2 = 0 + 37.5
5.5n + n2 = 0 + 37.5

Combine like terms: 0 + 37.5 = 37.5
5.5n + n2 = 37.5

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 = 37.5 + 7.5625

Reorder the terms:
7.5625 + 5.5n + n2 = 37.5 + 7.5625

Combine like terms: 37.5 + 7.5625 = 45.0625
7.5625 + 5.5n + n2 = 45.0625

Factor a perfect square on the left side:
(n + 2.75)(n + 2.75) = 45.0625

Calculate the square root of the right side: 6.712860791

Break this problem into two subproblems by setting 
(n + 2.75) equal to 6.712860791 and -6.712860791.

Subproblem 1

n + 2.75 = 6.712860791 Simplifying n + 2.75 = 6.712860791 Reorder the terms: 2.75 + n = 6.712860791 Solving 2.75 + n = 6.712860791 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 = 6.712860791 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + n = 6.712860791 + -2.75 n = 6.712860791 + -2.75 Combine like terms: 6.712860791 + -2.75 = 3.962860791 n = 3.962860791 Simplifying n = 3.962860791

Subproblem 2

n + 2.75 = -6.712860791 Simplifying n + 2.75 = -6.712860791 Reorder the terms: 2.75 + n = -6.712860791 Solving 2.75 + n = -6.712860791 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 = -6.712860791 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + n = -6.712860791 + -2.75 n = -6.712860791 + -2.75 Combine like terms: -6.712860791 + -2.75 = -9.462860791 n = -9.462860791 Simplifying n = -9.462860791

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.962860791, -9.462860791}

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