24n^2-35n-18=0

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Solution for 24n^2-35n-18=0 equation:


Simplifying
24n2 + -35n + -18 = 0

Reorder the terms:
-18 + -35n + 24n2 = 0

Solving
-18 + -35n + 24n2 = 0

Solving for variable 'n'.

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

Divide each side by '24'.
-0.75 + -1.458333333n + n2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + -1.458333333n + 0.75 + n2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + -1.458333333n + n2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + -1.458333333n + n2 = 0 + 0.75
-1.458333333n + n2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
-1.458333333n + n2 = 0.75

The n term is -1.458333333n.  Take half its coefficient (-0.7291666665).
Square it (0.5316840275) and add it to both sides.

Add '0.5316840275' to each side of the equation.
-1.458333333n + 0.5316840275 + n2 = 0.75 + 0.5316840275

Reorder the terms:
0.5316840275 + -1.458333333n + n2 = 0.75 + 0.5316840275

Combine like terms: 0.75 + 0.5316840275 = 1.2816840275
0.5316840275 + -1.458333333n + n2 = 1.2816840275

Factor a perfect square on the left side:
(n + -0.7291666665)(n + -0.7291666665) = 1.2816840275

Calculate the square root of the right side: 1.132114847

Break this problem into two subproblems by setting 
(n + -0.7291666665) equal to 1.132114847 and -1.132114847.

Subproblem 1

n + -0.7291666665 = 1.132114847 Simplifying n + -0.7291666665 = 1.132114847 Reorder the terms: -0.7291666665 + n = 1.132114847 Solving -0.7291666665 + n = 1.132114847 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.7291666665' to each side of the equation. -0.7291666665 + 0.7291666665 + n = 1.132114847 + 0.7291666665 Combine like terms: -0.7291666665 + 0.7291666665 = 0.0000000000 0.0000000000 + n = 1.132114847 + 0.7291666665 n = 1.132114847 + 0.7291666665 Combine like terms: 1.132114847 + 0.7291666665 = 1.8612815135 n = 1.8612815135 Simplifying n = 1.8612815135

Subproblem 2

n + -0.7291666665 = -1.132114847 Simplifying n + -0.7291666665 = -1.132114847 Reorder the terms: -0.7291666665 + n = -1.132114847 Solving -0.7291666665 + n = -1.132114847 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.7291666665' to each side of the equation. -0.7291666665 + 0.7291666665 + n = -1.132114847 + 0.7291666665 Combine like terms: -0.7291666665 + 0.7291666665 = 0.0000000000 0.0000000000 + n = -1.132114847 + 0.7291666665 n = -1.132114847 + 0.7291666665 Combine like terms: -1.132114847 + 0.7291666665 = -0.4029481805 n = -0.4029481805 Simplifying n = -0.4029481805

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.8612815135, -0.4029481805}

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