10n^2+12n-21=0

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Solution for 10n^2+12n-21=0 equation:


Simplifying
10n2 + 12n + -21 = 0

Reorder the terms:
-21 + 12n + 10n2 = 0

Solving
-21 + 12n + 10n2 = 0

Solving for variable 'n'.

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

Divide each side by '10'.
-2.1 + 1.2n + n2 = 0

Move the constant term to the right:

Add '2.1' to each side of the equation.
-2.1 + 1.2n + 2.1 + n2 = 0 + 2.1

Reorder the terms:
-2.1 + 2.1 + 1.2n + n2 = 0 + 2.1

Combine like terms: -2.1 + 2.1 = 0.0
0.0 + 1.2n + n2 = 0 + 2.1
1.2n + n2 = 0 + 2.1

Combine like terms: 0 + 2.1 = 2.1
1.2n + n2 = 2.1

The n term is 1.2n.  Take half its coefficient (0.6).
Square it (0.36) and add it to both sides.

Add '0.36' to each side of the equation.
1.2n + 0.36 + n2 = 2.1 + 0.36

Reorder the terms:
0.36 + 1.2n + n2 = 2.1 + 0.36

Combine like terms: 2.1 + 0.36 = 2.46
0.36 + 1.2n + n2 = 2.46

Factor a perfect square on the left side:
(n + 0.6)(n + 0.6) = 2.46

Calculate the square root of the right side: 1.568438714

Break this problem into two subproblems by setting 
(n + 0.6) equal to 1.568438714 and -1.568438714.

Subproblem 1

n + 0.6 = 1.568438714 Simplifying n + 0.6 = 1.568438714 Reorder the terms: 0.6 + n = 1.568438714 Solving 0.6 + n = 1.568438714 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + n = 1.568438714 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + n = 1.568438714 + -0.6 n = 1.568438714 + -0.6 Combine like terms: 1.568438714 + -0.6 = 0.968438714 n = 0.968438714 Simplifying n = 0.968438714

Subproblem 2

n + 0.6 = -1.568438714 Simplifying n + 0.6 = -1.568438714 Reorder the terms: 0.6 + n = -1.568438714 Solving 0.6 + n = -1.568438714 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + n = -1.568438714 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + n = -1.568438714 + -0.6 n = -1.568438714 + -0.6 Combine like terms: -1.568438714 + -0.6 = -2.168438714 n = -2.168438714 Simplifying n = -2.168438714

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.968438714, -2.168438714}

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