n^2-6n-51=3

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Solution for n^2-6n-51=3 equation:


Simplifying
n2 + -6n + -51 = 3

Reorder the terms:
-51 + -6n + n2 = 3

Solving
-51 + -6n + n2 = 3

Solving for variable 'n'.

Reorder the terms:
-51 + -3 + -6n + n2 = 3 + -3

Combine like terms: -51 + -3 = -54
-54 + -6n + n2 = 3 + -3

Combine like terms: 3 + -3 = 0
-54 + -6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + -6n + 54 + n2 = 0 + 54

Reorder the terms:
-54 + 54 + -6n + n2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + -6n + n2 = 0 + 54
-6n + n2 = 0 + 54

Combine like terms: 0 + 54 = 54
-6n + n2 = 54

The n term is -6n.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6n + 9 + n2 = 54 + 9

Reorder the terms:
9 + -6n + n2 = 54 + 9

Combine like terms: 54 + 9 = 63
9 + -6n + n2 = 63

Factor a perfect square on the left side:
(n + -3)(n + -3) = 63

Calculate the square root of the right side: 7.937253933

Break this problem into two subproblems by setting 
(n + -3) equal to 7.937253933 and -7.937253933.

Subproblem 1

n + -3 = 7.937253933 Simplifying n + -3 = 7.937253933 Reorder the terms: -3 + n = 7.937253933 Solving -3 + n = 7.937253933 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 7.937253933 + 3 Combine like terms: -3 + 3 = 0 0 + n = 7.937253933 + 3 n = 7.937253933 + 3 Combine like terms: 7.937253933 + 3 = 10.937253933 n = 10.937253933 Simplifying n = 10.937253933

Subproblem 2

n + -3 = -7.937253933 Simplifying n + -3 = -7.937253933 Reorder the terms: -3 + n = -7.937253933 Solving -3 + n = -7.937253933 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -7.937253933 + 3 Combine like terms: -3 + 3 = 0 0 + n = -7.937253933 + 3 n = -7.937253933 + 3 Combine like terms: -7.937253933 + 3 = -4.937253933 n = -4.937253933 Simplifying n = -4.937253933

Solution

The solution to the problem is based on the solutions from the subproblems. n = {10.937253933, -4.937253933}

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