4n^2-17n=19

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Solution for 4n^2-17n=19 equation:


Simplifying
4n2 + -17n = 19

Reorder the terms:
-17n + 4n2 = 19

Solving
-17n + 4n2 = 19

Solving for variable 'n'.

Reorder the terms:
-19 + -17n + 4n2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + -17n + 4n2 = 0

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

Divide each side by '4'.
-4.75 + -4.25n + n2 = 0

Move the constant term to the right:

Add '4.75' to each side of the equation.
-4.75 + -4.25n + 4.75 + n2 = 0 + 4.75

Reorder the terms:
-4.75 + 4.75 + -4.25n + n2 = 0 + 4.75

Combine like terms: -4.75 + 4.75 = 0.00
0.00 + -4.25n + n2 = 0 + 4.75
-4.25n + n2 = 0 + 4.75

Combine like terms: 0 + 4.75 = 4.75
-4.25n + n2 = 4.75

The n term is -4.25n.  Take half its coefficient (-2.125).
Square it (4.515625) and add it to both sides.

Add '4.515625' to each side of the equation.
-4.25n + 4.515625 + n2 = 4.75 + 4.515625

Reorder the terms:
4.515625 + -4.25n + n2 = 4.75 + 4.515625

Combine like terms: 4.75 + 4.515625 = 9.265625
4.515625 + -4.25n + n2 = 9.265625

Factor a perfect square on the left side:
(n + -2.125)(n + -2.125) = 9.265625

Calculate the square root of the right side: 3.043948915

Break this problem into two subproblems by setting 
(n + -2.125) equal to 3.043948915 and -3.043948915.

Subproblem 1

n + -2.125 = 3.043948915 Simplifying n + -2.125 = 3.043948915 Reorder the terms: -2.125 + n = 3.043948915 Solving -2.125 + n = 3.043948915 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.125' to each side of the equation. -2.125 + 2.125 + n = 3.043948915 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + n = 3.043948915 + 2.125 n = 3.043948915 + 2.125 Combine like terms: 3.043948915 + 2.125 = 5.168948915 n = 5.168948915 Simplifying n = 5.168948915

Subproblem 2

n + -2.125 = -3.043948915 Simplifying n + -2.125 = -3.043948915 Reorder the terms: -2.125 + n = -3.043948915 Solving -2.125 + n = -3.043948915 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.125' to each side of the equation. -2.125 + 2.125 + n = -3.043948915 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + n = -3.043948915 + 2.125 n = -3.043948915 + 2.125 Combine like terms: -3.043948915 + 2.125 = -0.918948915 n = -0.918948915 Simplifying n = -0.918948915

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.168948915, -0.918948915}

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