2n^2+3n-21=0

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Solution for 2n^2+3n-21=0 equation:


Simplifying
2n2 + 3n + -21 = 0

Reorder the terms:
-21 + 3n + 2n2 = 0

Solving
-21 + 3n + 2n2 = 0

Solving for variable 'n'.

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

Divide each side by '2'.
-10.5 + 1.5n + n2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + 1.5n + 10.5 + n2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + 1.5n + n2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + 1.5n + n2 = 0 + 10.5
1.5n + n2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
1.5n + n2 = 10.5

The n term is 1.5n.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5n + 0.5625 + n2 = 10.5 + 0.5625

Reorder the terms:
0.5625 + 1.5n + n2 = 10.5 + 0.5625

Combine like terms: 10.5 + 0.5625 = 11.0625
0.5625 + 1.5n + n2 = 11.0625

Factor a perfect square on the left side:
(n + 0.75)(n + 0.75) = 11.0625

Calculate the square root of the right side: 3.326033674

Break this problem into two subproblems by setting 
(n + 0.75) equal to 3.326033674 and -3.326033674.

Subproblem 1

n + 0.75 = 3.326033674 Simplifying n + 0.75 = 3.326033674 Reorder the terms: 0.75 + n = 3.326033674 Solving 0.75 + n = 3.326033674 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = 3.326033674 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = 3.326033674 + -0.75 n = 3.326033674 + -0.75 Combine like terms: 3.326033674 + -0.75 = 2.576033674 n = 2.576033674 Simplifying n = 2.576033674

Subproblem 2

n + 0.75 = -3.326033674 Simplifying n + 0.75 = -3.326033674 Reorder the terms: 0.75 + n = -3.326033674 Solving 0.75 + n = -3.326033674 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = -3.326033674 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = -3.326033674 + -0.75 n = -3.326033674 + -0.75 Combine like terms: -3.326033674 + -0.75 = -4.076033674 n = -4.076033674 Simplifying n = -4.076033674

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.576033674, -4.076033674}

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