n+n^2=75

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Solution for n+n^2=75 equation:


Simplifying
n + n2 = 75

Solving
n + n2 = 75

Solving for variable 'n'.

Reorder the terms:
-75 + n + n2 = 75 + -75

Combine like terms: 75 + -75 = 0
-75 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '75' to each side of the equation.
-75 + n + 75 + n2 = 0 + 75

Reorder the terms:
-75 + 75 + n + n2 = 0 + 75

Combine like terms: -75 + 75 = 0
0 + n + n2 = 0 + 75
n + n2 = 0 + 75

Combine like terms: 0 + 75 = 75
n + n2 = 75

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
n + 0.25 + n2 = 75 + 0.25

Reorder the terms:
0.25 + n + n2 = 75 + 0.25

Combine like terms: 75 + 0.25 = 75.25
0.25 + n + n2 = 75.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 75.25

Calculate the square root of the right side: 8.674675786

Break this problem into two subproblems by setting 
(n + 0.5) equal to 8.674675786 and -8.674675786.

Subproblem 1

n + 0.5 = 8.674675786 Simplifying n + 0.5 = 8.674675786 Reorder the terms: 0.5 + n = 8.674675786 Solving 0.5 + n = 8.674675786 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 8.674675786 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 8.674675786 + -0.5 n = 8.674675786 + -0.5 Combine like terms: 8.674675786 + -0.5 = 8.174675786 n = 8.174675786 Simplifying n = 8.174675786

Subproblem 2

n + 0.5 = -8.674675786 Simplifying n + 0.5 = -8.674675786 Reorder the terms: 0.5 + n = -8.674675786 Solving 0.5 + n = -8.674675786 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -8.674675786 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -8.674675786 + -0.5 n = -8.674675786 + -0.5 Combine like terms: -8.674675786 + -0.5 = -9.174675786 n = -9.174675786 Simplifying n = -9.174675786

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8.174675786, -9.174675786}

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