72+n^2=30n

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Solution for 72+n^2=30n equation:


Simplifying
72 + n2 = 30n

Solving
72 + n2 = 30n

Solving for variable 'n'.

Reorder the terms:
72 + -30n + n2 = 30n + -30n

Combine like terms: 30n + -30n = 0
72 + -30n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-72' to each side of the equation.
72 + -30n + -72 + n2 = 0 + -72

Reorder the terms:
72 + -72 + -30n + n2 = 0 + -72

Combine like terms: 72 + -72 = 0
0 + -30n + n2 = 0 + -72
-30n + n2 = 0 + -72

Combine like terms: 0 + -72 = -72
-30n + n2 = -72

The n term is -30n.  Take half its coefficient (-15).
Square it (225) and add it to both sides.

Add '225' to each side of the equation.
-30n + 225 + n2 = -72 + 225

Reorder the terms:
225 + -30n + n2 = -72 + 225

Combine like terms: -72 + 225 = 153
225 + -30n + n2 = 153

Factor a perfect square on the left side:
(n + -15)(n + -15) = 153

Calculate the square root of the right side: 12.369316877

Break this problem into two subproblems by setting 
(n + -15) equal to 12.369316877 and -12.369316877.

Subproblem 1

n + -15 = 12.369316877 Simplifying n + -15 = 12.369316877 Reorder the terms: -15 + n = 12.369316877 Solving -15 + n = 12.369316877 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '15' to each side of the equation. -15 + 15 + n = 12.369316877 + 15 Combine like terms: -15 + 15 = 0 0 + n = 12.369316877 + 15 n = 12.369316877 + 15 Combine like terms: 12.369316877 + 15 = 27.369316877 n = 27.369316877 Simplifying n = 27.369316877

Subproblem 2

n + -15 = -12.369316877 Simplifying n + -15 = -12.369316877 Reorder the terms: -15 + n = -12.369316877 Solving -15 + n = -12.369316877 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '15' to each side of the equation. -15 + 15 + n = -12.369316877 + 15 Combine like terms: -15 + 15 = 0 0 + n = -12.369316877 + 15 n = -12.369316877 + 15 Combine like terms: -12.369316877 + 15 = 2.630683123 n = 2.630683123 Simplifying n = 2.630683123

Solution

The solution to the problem is based on the solutions from the subproblems. n = {27.369316877, 2.630683123}

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