9n^2+30n-10=0

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Solution for 9n^2+30n-10=0 equation:


Simplifying
9n2 + 30n + -10 = 0

Reorder the terms:
-10 + 30n + 9n2 = 0

Solving
-10 + 30n + 9n2 = 0

Solving for variable 'n'.

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

Divide each side by '9'.
-1.111111111 + 3.333333333n + n2 = 0

Move the constant term to the right:

Add '1.111111111' to each side of the equation.
-1.111111111 + 3.333333333n + 1.111111111 + n2 = 0 + 1.111111111

Reorder the terms:
-1.111111111 + 1.111111111 + 3.333333333n + n2 = 0 + 1.111111111

Combine like terms: -1.111111111 + 1.111111111 = 0.000000000
0.000000000 + 3.333333333n + n2 = 0 + 1.111111111
3.333333333n + n2 = 0 + 1.111111111

Combine like terms: 0 + 1.111111111 = 1.111111111
3.333333333n + n2 = 1.111111111

The n term is 3.333333333n.  Take half its coefficient (1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
3.333333333n + 2.777777779 + n2 = 1.111111111 + 2.777777779

Reorder the terms:
2.777777779 + 3.333333333n + n2 = 1.111111111 + 2.777777779

Combine like terms: 1.111111111 + 2.777777779 = 3.88888889
2.777777779 + 3.333333333n + n2 = 3.88888889

Factor a perfect square on the left side:
(n + 1.666666667)(n + 1.666666667) = 3.88888889

Calculate the square root of the right side: 1.972026595

Break this problem into two subproblems by setting 
(n + 1.666666667) equal to 1.972026595 and -1.972026595.

Subproblem 1

n + 1.666666667 = 1.972026595 Simplifying n + 1.666666667 = 1.972026595 Reorder the terms: 1.666666667 + n = 1.972026595 Solving 1.666666667 + n = 1.972026595 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + n = 1.972026595 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + n = 1.972026595 + -1.666666667 n = 1.972026595 + -1.666666667 Combine like terms: 1.972026595 + -1.666666667 = 0.305359928 n = 0.305359928 Simplifying n = 0.305359928

Subproblem 2

n + 1.666666667 = -1.972026595 Simplifying n + 1.666666667 = -1.972026595 Reorder the terms: 1.666666667 + n = -1.972026595 Solving 1.666666667 + n = -1.972026595 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + n = -1.972026595 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + n = -1.972026595 + -1.666666667 n = -1.972026595 + -1.666666667 Combine like terms: -1.972026595 + -1.666666667 = -3.638693262 n = -3.638693262 Simplifying n = -3.638693262

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.305359928, -3.638693262}

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