2n^2+5n=24

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Solution for 2n^2+5n=24 equation:


Simplifying
2n2 + 5n = 24

Reorder the terms:
5n + 2n2 = 24

Solving
5n + 2n2 = 24

Solving for variable 'n'.

Reorder the terms:
-24 + 5n + 2n2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + 5n + 2n2 = 0

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

Divide each side by '2'.
-12 + 2.5n + n2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 2.5n + 12 + n2 = 0 + 12

Reorder the terms:
-12 + 12 + 2.5n + n2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 2.5n + n2 = 0 + 12
2.5n + n2 = 0 + 12

Combine like terms: 0 + 12 = 12
2.5n + n2 = 12

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 = 12 + 1.5625

Reorder the terms:
1.5625 + 2.5n + n2 = 12 + 1.5625

Combine like terms: 12 + 1.5625 = 13.5625
1.5625 + 2.5n + n2 = 13.5625

Factor a perfect square on the left side:
(n + 1.25)(n + 1.25) = 13.5625

Calculate the square root of the right side: 3.682729966

Break this problem into two subproblems by setting 
(n + 1.25) equal to 3.682729966 and -3.682729966.

Subproblem 1

n + 1.25 = 3.682729966 Simplifying n + 1.25 = 3.682729966 Reorder the terms: 1.25 + n = 3.682729966 Solving 1.25 + n = 3.682729966 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 = 3.682729966 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 3.682729966 + -1.25 n = 3.682729966 + -1.25 Combine like terms: 3.682729966 + -1.25 = 2.432729966 n = 2.432729966 Simplifying n = 2.432729966

Subproblem 2

n + 1.25 = -3.682729966 Simplifying n + 1.25 = -3.682729966 Reorder the terms: 1.25 + n = -3.682729966 Solving 1.25 + n = -3.682729966 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 = -3.682729966 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -3.682729966 + -1.25 n = -3.682729966 + -1.25 Combine like terms: -3.682729966 + -1.25 = -4.932729966 n = -4.932729966 Simplifying n = -4.932729966

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.432729966, -4.932729966}

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