2n^2+5n=9

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Solution for 2n^2+5n=9 equation:


Simplifying
2n2 + 5n = 9

Reorder the terms:
5n + 2n2 = 9

Solving
5n + 2n2 = 9

Solving for variable 'n'.

Reorder the terms:
-9 + 5n + 2n2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 5n + 2n2 = 0

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

Divide each side by '2'.
-4.5 + 2.5n + n2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 2.5n + 4.5 + n2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 2.5n + n2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 2.5n + n2 = 0 + 4.5
2.5n + n2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
2.5n + n2 = 4.5

The n term is 2.5n.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5n + 1.5625 + n2 = 4.5 + 1.5625

Reorder the terms:
1.5625 + 2.5n + n2 = 4.5 + 1.5625

Combine like terms: 4.5 + 1.5625 = 6.0625
1.5625 + 2.5n + n2 = 6.0625

Factor a perfect square on the left side:
(n + 1.25)(n + 1.25) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(n + 1.25) equal to 2.46221445 and -2.46221445.

Subproblem 1

n + 1.25 = 2.46221445 Simplifying n + 1.25 = 2.46221445 Reorder the terms: 1.25 + n = 2.46221445 Solving 1.25 + n = 2.46221445 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = 2.46221445 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 2.46221445 + -1.25 n = 2.46221445 + -1.25 Combine like terms: 2.46221445 + -1.25 = 1.21221445 n = 1.21221445 Simplifying n = 1.21221445

Subproblem 2

n + 1.25 = -2.46221445 Simplifying n + 1.25 = -2.46221445 Reorder the terms: 1.25 + n = -2.46221445 Solving 1.25 + n = -2.46221445 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = -2.46221445 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -2.46221445 + -1.25 n = -2.46221445 + -1.25 Combine like terms: -2.46221445 + -1.25 = -3.71221445 n = -3.71221445 Simplifying n = -3.71221445

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.21221445, -3.71221445}

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