21n^2+13n-30=0

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Solution for 21n^2+13n-30=0 equation:


Simplifying
21n2 + 13n + -30 = 0

Reorder the terms:
-30 + 13n + 21n2 = 0

Solving
-30 + 13n + 21n2 = 0

Solving for variable 'n'.

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

Divide each side by '21'.
-1.428571429 + 0.619047619n + n2 = 0

Move the constant term to the right:

Add '1.428571429' to each side of the equation.
-1.428571429 + 0.619047619n + 1.428571429 + n2 = 0 + 1.428571429

Reorder the terms:
-1.428571429 + 1.428571429 + 0.619047619n + n2 = 0 + 1.428571429

Combine like terms: -1.428571429 + 1.428571429 = 0.000000000
0.000000000 + 0.619047619n + n2 = 0 + 1.428571429
0.619047619n + n2 = 0 + 1.428571429

Combine like terms: 0 + 1.428571429 = 1.428571429
0.619047619n + n2 = 1.428571429

The n term is 0.619047619n.  Take half its coefficient (0.3095238095).
Square it (0.09580498865) and add it to both sides.

Add '0.09580498865' to each side of the equation.
0.619047619n + 0.09580498865 + n2 = 1.428571429 + 0.09580498865

Reorder the terms:
0.09580498865 + 0.619047619n + n2 = 1.428571429 + 0.09580498865

Combine like terms: 1.428571429 + 0.09580498865 = 1.52437641765
0.09580498865 + 0.619047619n + n2 = 1.52437641765

Factor a perfect square on the left side:
(n + 0.3095238095)(n + 0.3095238095) = 1.52437641765

Calculate the square root of the right side: 1.234656397

Break this problem into two subproblems by setting 
(n + 0.3095238095) equal to 1.234656397 and -1.234656397.

Subproblem 1

n + 0.3095238095 = 1.234656397 Simplifying n + 0.3095238095 = 1.234656397 Reorder the terms: 0.3095238095 + n = 1.234656397 Solving 0.3095238095 + n = 1.234656397 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3095238095' to each side of the equation. 0.3095238095 + -0.3095238095 + n = 1.234656397 + -0.3095238095 Combine like terms: 0.3095238095 + -0.3095238095 = 0.0000000000 0.0000000000 + n = 1.234656397 + -0.3095238095 n = 1.234656397 + -0.3095238095 Combine like terms: 1.234656397 + -0.3095238095 = 0.9251325875 n = 0.9251325875 Simplifying n = 0.9251325875

Subproblem 2

n + 0.3095238095 = -1.234656397 Simplifying n + 0.3095238095 = -1.234656397 Reorder the terms: 0.3095238095 + n = -1.234656397 Solving 0.3095238095 + n = -1.234656397 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3095238095' to each side of the equation. 0.3095238095 + -0.3095238095 + n = -1.234656397 + -0.3095238095 Combine like terms: 0.3095238095 + -0.3095238095 = 0.0000000000 0.0000000000 + n = -1.234656397 + -0.3095238095 n = -1.234656397 + -0.3095238095 Combine like terms: -1.234656397 + -0.3095238095 = -1.5441802065 n = -1.5441802065 Simplifying n = -1.5441802065

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.9251325875, -1.5441802065}

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