15n^2-n=72

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Solution for 15n^2-n=72 equation:


Simplifying
15n2 + -1n = 72

Reorder the terms:
-1n + 15n2 = 72

Solving
-1n + 15n2 = 72

Solving for variable 'n'.

Reorder the terms:
-72 + -1n + 15n2 = 72 + -72

Combine like terms: 72 + -72 = 0
-72 + -1n + 15n2 = 0

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

Divide each side by '15'.
-4.8 + -0.06666666667n + n2 = 0

Move the constant term to the right:

Add '4.8' to each side of the equation.
-4.8 + -0.06666666667n + 4.8 + n2 = 0 + 4.8

Reorder the terms:
-4.8 + 4.8 + -0.06666666667n + n2 = 0 + 4.8

Combine like terms: -4.8 + 4.8 = 0.0
0.0 + -0.06666666667n + n2 = 0 + 4.8
-0.06666666667n + n2 = 0 + 4.8

Combine like terms: 0 + 4.8 = 4.8
-0.06666666667n + n2 = 4.8

The n term is -0.06666666667n.  Take half its coefficient (-0.03333333334).
Square it (0.001111111112) and add it to both sides.

Add '0.001111111112' to each side of the equation.
-0.06666666667n + 0.001111111112 + n2 = 4.8 + 0.001111111112

Reorder the terms:
0.001111111112 + -0.06666666667n + n2 = 4.8 + 0.001111111112

Combine like terms: 4.8 + 0.001111111112 = 4.801111111112
0.001111111112 + -0.06666666667n + n2 = 4.801111111112

Factor a perfect square on the left side:
(n + -0.03333333334)(n + -0.03333333334) = 4.801111111112

Calculate the square root of the right side: 2.191143791

Break this problem into two subproblems by setting 
(n + -0.03333333334) equal to 2.191143791 and -2.191143791.

Subproblem 1

n + -0.03333333334 = 2.191143791 Simplifying n + -0.03333333334 = 2.191143791 Reorder the terms: -0.03333333334 + n = 2.191143791 Solving -0.03333333334 + n = 2.191143791 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.03333333334' to each side of the equation. -0.03333333334 + 0.03333333334 + n = 2.191143791 + 0.03333333334 Combine like terms: -0.03333333334 + 0.03333333334 = 0.00000000000 0.00000000000 + n = 2.191143791 + 0.03333333334 n = 2.191143791 + 0.03333333334 Combine like terms: 2.191143791 + 0.03333333334 = 2.22447712434 n = 2.22447712434 Simplifying n = 2.22447712434

Subproblem 2

n + -0.03333333334 = -2.191143791 Simplifying n + -0.03333333334 = -2.191143791 Reorder the terms: -0.03333333334 + n = -2.191143791 Solving -0.03333333334 + n = -2.191143791 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.03333333334' to each side of the equation. -0.03333333334 + 0.03333333334 + n = -2.191143791 + 0.03333333334 Combine like terms: -0.03333333334 + 0.03333333334 = 0.00000000000 0.00000000000 + n = -2.191143791 + 0.03333333334 n = -2.191143791 + 0.03333333334 Combine like terms: -2.191143791 + 0.03333333334 = -2.15781045766 n = -2.15781045766 Simplifying n = -2.15781045766

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.22447712434, -2.15781045766}

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