n^2+9n-3=0

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Solution for n^2+9n-3=0 equation:


Simplifying
n2 + 9n + -3 = 0

Reorder the terms:
-3 + 9n + n2 = 0

Solving
-3 + 9n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 9n + 3 + n2 = 0 + 3

Reorder the terms:
-3 + 3 + 9n + n2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 9n + n2 = 0 + 3
9n + n2 = 0 + 3

Combine like terms: 0 + 3 = 3
9n + n2 = 3

The n term is 9n.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9n + 20.25 + n2 = 3 + 20.25

Reorder the terms:
20.25 + 9n + n2 = 3 + 20.25

Combine like terms: 3 + 20.25 = 23.25
20.25 + 9n + n2 = 23.25

Factor a perfect square on the left side:
(n + 4.5)(n + 4.5) = 23.25

Calculate the square root of the right side: 4.82182538

Break this problem into two subproblems by setting 
(n + 4.5) equal to 4.82182538 and -4.82182538.

Subproblem 1

n + 4.5 = 4.82182538 Simplifying n + 4.5 = 4.82182538 Reorder the terms: 4.5 + n = 4.82182538 Solving 4.5 + n = 4.82182538 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = 4.82182538 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = 4.82182538 + -4.5 n = 4.82182538 + -4.5 Combine like terms: 4.82182538 + -4.5 = 0.32182538 n = 0.32182538 Simplifying n = 0.32182538

Subproblem 2

n + 4.5 = -4.82182538 Simplifying n + 4.5 = -4.82182538 Reorder the terms: 4.5 + n = -4.82182538 Solving 4.5 + n = -4.82182538 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = -4.82182538 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = -4.82182538 + -4.5 n = -4.82182538 + -4.5 Combine like terms: -4.82182538 + -4.5 = -9.32182538 n = -9.32182538 Simplifying n = -9.32182538

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.32182538, -9.32182538}

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