4n^2+19n-18=0

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Solution for 4n^2+19n-18=0 equation:


Simplifying
4n2 + 19n + -18 = 0

Reorder the terms:
-18 + 19n + 4n2 = 0

Solving
-18 + 19n + 4n2 = 0

Solving for variable 'n'.

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

Divide each side by '4'.
-4.5 + 4.75n + n2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 4.75n + 4.5 + n2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 4.75n + n2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 4.75n + n2 = 0 + 4.5
4.75n + n2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
4.75n + n2 = 4.5

The n term is 4.75n.  Take half its coefficient (2.375).
Square it (5.640625) and add it to both sides.

Add '5.640625' to each side of the equation.
4.75n + 5.640625 + n2 = 4.5 + 5.640625

Reorder the terms:
5.640625 + 4.75n + n2 = 4.5 + 5.640625

Combine like terms: 4.5 + 5.640625 = 10.140625
5.640625 + 4.75n + n2 = 10.140625

Factor a perfect square on the left side:
(n + 2.375)(n + 2.375) = 10.140625

Calculate the square root of the right side: 3.184434801

Break this problem into two subproblems by setting 
(n + 2.375) equal to 3.184434801 and -3.184434801.

Subproblem 1

n + 2.375 = 3.184434801 Simplifying n + 2.375 = 3.184434801 Reorder the terms: 2.375 + n = 3.184434801 Solving 2.375 + n = 3.184434801 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + n = 3.184434801 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + n = 3.184434801 + -2.375 n = 3.184434801 + -2.375 Combine like terms: 3.184434801 + -2.375 = 0.809434801 n = 0.809434801 Simplifying n = 0.809434801

Subproblem 2

n + 2.375 = -3.184434801 Simplifying n + 2.375 = -3.184434801 Reorder the terms: 2.375 + n = -3.184434801 Solving 2.375 + n = -3.184434801 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + n = -3.184434801 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + n = -3.184434801 + -2.375 n = -3.184434801 + -2.375 Combine like terms: -3.184434801 + -2.375 = -5.559434801 n = -5.559434801 Simplifying n = -5.559434801

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.809434801, -5.559434801}

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