-8(n)=n^2-3

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Solution for -8(n)=n^2-3 equation:


Simplifying
-8(n) = n2 + -3

Reorder the terms:
-8n = -3 + n2

Solving
-8n = -3 + n2

Solving for variable 'n'.

Reorder the terms:
3 + -8n + -1n2 = -3 + n2 + 3 + -1n2

Reorder the terms:
3 + -8n + -1n2 = -3 + 3 + n2 + -1n2

Combine like terms: -3 + 3 = 0
3 + -8n + -1n2 = 0 + n2 + -1n2
3 + -8n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
3 + -8n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-3 + 8n + n2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 8n + 3 + n2 = 0 + 3

Reorder the terms:
-3 + 3 + 8n + n2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 8n + n2 = 0 + 3
8n + n2 = 0 + 3

Combine like terms: 0 + 3 = 3
8n + n2 = 3

The n term is 8n.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8n + 16 + n2 = 3 + 16

Reorder the terms:
16 + 8n + n2 = 3 + 16

Combine like terms: 3 + 16 = 19
16 + 8n + n2 = 19

Factor a perfect square on the left side:
(n + 4)(n + 4) = 19

Calculate the square root of the right side: 4.358898944

Break this problem into two subproblems by setting 
(n + 4) equal to 4.358898944 and -4.358898944.

Subproblem 1

n + 4 = 4.358898944 Simplifying n + 4 = 4.358898944 Reorder the terms: 4 + n = 4.358898944 Solving 4 + n = 4.358898944 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + n = 4.358898944 + -4 Combine like terms: 4 + -4 = 0 0 + n = 4.358898944 + -4 n = 4.358898944 + -4 Combine like terms: 4.358898944 + -4 = 0.358898944 n = 0.358898944 Simplifying n = 0.358898944

Subproblem 2

n + 4 = -4.358898944 Simplifying n + 4 = -4.358898944 Reorder the terms: 4 + n = -4.358898944 Solving 4 + n = -4.358898944 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + n = -4.358898944 + -4 Combine like terms: 4 + -4 = 0 0 + n = -4.358898944 + -4 n = -4.358898944 + -4 Combine like terms: -4.358898944 + -4 = -8.358898944 n = -8.358898944 Simplifying n = -8.358898944

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.358898944, -8.358898944}

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