3n^2+5n-30=0

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Solution for 3n^2+5n-30=0 equation:


Simplifying
3n2 + 5n + -30 = 0

Reorder the terms:
-30 + 5n + 3n2 = 0

Solving
-30 + 5n + 3n2 = 0

Solving for variable 'n'.

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

Divide each side by '3'.
-10 + 1.666666667n + n2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 1.666666667n + 10 + n2 = 0 + 10

Reorder the terms:
-10 + 10 + 1.666666667n + n2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 1.666666667n + n2 = 0 + 10
1.666666667n + n2 = 0 + 10

Combine like terms: 0 + 10 = 10
1.666666667n + n2 = 10

The n term is 1.666666667n.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667n + 0.6944444447 + n2 = 10 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667n + n2 = 10 + 0.6944444447

Combine like terms: 10 + 0.6944444447 = 10.6944444447
0.6944444447 + 1.666666667n + n2 = 10.6944444447

Factor a perfect square on the left side:
(n + 0.8333333335)(n + 0.8333333335) = 10.6944444447

Calculate the square root of the right side: 3.270236145

Break this problem into two subproblems by setting 
(n + 0.8333333335) equal to 3.270236145 and -3.270236145.

Subproblem 1

n + 0.8333333335 = 3.270236145 Simplifying n + 0.8333333335 = 3.270236145 Reorder the terms: 0.8333333335 + n = 3.270236145 Solving 0.8333333335 + n = 3.270236145 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + n = 3.270236145 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + n = 3.270236145 + -0.8333333335 n = 3.270236145 + -0.8333333335 Combine like terms: 3.270236145 + -0.8333333335 = 2.4369028115 n = 2.4369028115 Simplifying n = 2.4369028115

Subproblem 2

n + 0.8333333335 = -3.270236145 Simplifying n + 0.8333333335 = -3.270236145 Reorder the terms: 0.8333333335 + n = -3.270236145 Solving 0.8333333335 + n = -3.270236145 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + n = -3.270236145 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + n = -3.270236145 + -0.8333333335 n = -3.270236145 + -0.8333333335 Combine like terms: -3.270236145 + -0.8333333335 = -4.1035694785 n = -4.1035694785 Simplifying n = -4.1035694785

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.4369028115, -4.1035694785}

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