3n*5=3*1-n^2

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Solution for 3n*5=3*1-n^2 equation:


Simplifying
3n * 5 = 3 * 1 + -1n2

Reorder the terms for easier multiplication:
3 * 5n = 3 * 1 + -1n2

Multiply 3 * 5
15n = 3 * 1 + -1n2

Multiply 3 * 1
15n = 3 + -1n2

Solving
15n = 3 + -1n2

Solving for variable 'n'.

Reorder the terms:
-3 + 15n + n2 = 3 + -1n2 + -3 + n2

Reorder the terms:
-3 + 15n + n2 = 3 + -3 + -1n2 + n2

Combine like terms: 3 + -3 = 0
-3 + 15n + n2 = 0 + -1n2 + n2
-3 + 15n + n2 = -1n2 + n2

Combine like terms: -1n2 + n2 = 0
-3 + 15n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 15n + 3 + n2 = 0 + 3

Reorder the terms:
-3 + 3 + 15n + n2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 15n + n2 = 0 + 3
15n + n2 = 0 + 3

Combine like terms: 0 + 3 = 3
15n + n2 = 3

The n term is 15n.  Take half its coefficient (7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
15n + 56.25 + n2 = 3 + 56.25

Reorder the terms:
56.25 + 15n + n2 = 3 + 56.25

Combine like terms: 3 + 56.25 = 59.25
56.25 + 15n + n2 = 59.25

Factor a perfect square on the left side:
(n + 7.5)(n + 7.5) = 59.25

Calculate the square root of the right side: 7.697402159

Break this problem into two subproblems by setting 
(n + 7.5) equal to 7.697402159 and -7.697402159.

Subproblem 1

n + 7.5 = 7.697402159 Simplifying n + 7.5 = 7.697402159 Reorder the terms: 7.5 + n = 7.697402159 Solving 7.5 + n = 7.697402159 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + n = 7.697402159 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + n = 7.697402159 + -7.5 n = 7.697402159 + -7.5 Combine like terms: 7.697402159 + -7.5 = 0.197402159 n = 0.197402159 Simplifying n = 0.197402159

Subproblem 2

n + 7.5 = -7.697402159 Simplifying n + 7.5 = -7.697402159 Reorder the terms: 7.5 + n = -7.697402159 Solving 7.5 + n = -7.697402159 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + n = -7.697402159 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + n = -7.697402159 + -7.5 n = -7.697402159 + -7.5 Combine like terms: -7.697402159 + -7.5 = -15.197402159 n = -15.197402159 Simplifying n = -15.197402159

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.197402159, -15.197402159}

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