3n^2+5n=4

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Solution for 3n^2+5n=4 equation:


Simplifying
3n2 + 5n = 4

Reorder the terms:
5n + 3n2 = 4

Solving
5n + 3n2 = 4

Solving for variable 'n'.

Reorder the terms:
-4 + 5n + 3n2 = 4 + -4

Combine like terms: 4 + -4 = 0
-4 + 5n + 3n2 = 0

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

Divide each side by '3'.
-1.333333333 + 1.666666667n + n2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + 1.666666667n + 1.333333333 + n2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + 1.666666667n + n2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + 1.666666667n + n2 = 0 + 1.333333333
1.666666667n + n2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
1.666666667n + n2 = 1.333333333

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 = 1.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667n + n2 = 1.333333333 + 0.6944444447

Combine like terms: 1.333333333 + 0.6944444447 = 2.0277777777
0.6944444447 + 1.666666667n + n2 = 2.0277777777

Factor a perfect square on the left side:
(n + 0.8333333335)(n + 0.8333333335) = 2.0277777777

Calculate the square root of the right side: 1.424000624

Break this problem into two subproblems by setting 
(n + 0.8333333335) equal to 1.424000624 and -1.424000624.

Subproblem 1

n + 0.8333333335 = 1.424000624 Simplifying n + 0.8333333335 = 1.424000624 Reorder the terms: 0.8333333335 + n = 1.424000624 Solving 0.8333333335 + n = 1.424000624 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 = 1.424000624 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + n = 1.424000624 + -0.8333333335 n = 1.424000624 + -0.8333333335 Combine like terms: 1.424000624 + -0.8333333335 = 0.5906672905 n = 0.5906672905 Simplifying n = 0.5906672905

Subproblem 2

n + 0.8333333335 = -1.424000624 Simplifying n + 0.8333333335 = -1.424000624 Reorder the terms: 0.8333333335 + n = -1.424000624 Solving 0.8333333335 + n = -1.424000624 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 = -1.424000624 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + n = -1.424000624 + -0.8333333335 n = -1.424000624 + -0.8333333335 Combine like terms: -1.424000624 + -0.8333333335 = -2.2573339575 n = -2.2573339575 Simplifying n = -2.2573339575

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.5906672905, -2.2573339575}

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