7n^2-2n-21=-5

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Solution for 7n^2-2n-21=-5 equation:


Simplifying
7n2 + -2n + -21 = -5

Reorder the terms:
-21 + -2n + 7n2 = -5

Solving
-21 + -2n + 7n2 = -5

Solving for variable 'n'.

Reorder the terms:
-21 + 5 + -2n + 7n2 = -5 + 5

Combine like terms: -21 + 5 = -16
-16 + -2n + 7n2 = -5 + 5

Combine like terms: -5 + 5 = 0
-16 + -2n + 7n2 = 0

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

Divide each side by '7'.
-2.285714286 + -0.2857142857n + n2 = 0

Move the constant term to the right:

Add '2.285714286' to each side of the equation.
-2.285714286 + -0.2857142857n + 2.285714286 + n2 = 0 + 2.285714286

Reorder the terms:
-2.285714286 + 2.285714286 + -0.2857142857n + n2 = 0 + 2.285714286

Combine like terms: -2.285714286 + 2.285714286 = 0.000000000
0.000000000 + -0.2857142857n + n2 = 0 + 2.285714286
-0.2857142857n + n2 = 0 + 2.285714286

Combine like terms: 0 + 2.285714286 = 2.285714286
-0.2857142857n + n2 = 2.285714286

The n term is -0.2857142857n.  Take half its coefficient (-0.1428571429).
Square it (0.02040816328) and add it to both sides.

Add '0.02040816328' to each side of the equation.
-0.2857142857n + 0.02040816328 + n2 = 2.285714286 + 0.02040816328

Reorder the terms:
0.02040816328 + -0.2857142857n + n2 = 2.285714286 + 0.02040816328

Combine like terms: 2.285714286 + 0.02040816328 = 2.30612244928
0.02040816328 + -0.2857142857n + n2 = 2.30612244928

Factor a perfect square on the left side:
(n + -0.1428571429)(n + -0.1428571429) = 2.30612244928

Calculate the square root of the right side: 1.518592259

Break this problem into two subproblems by setting 
(n + -0.1428571429) equal to 1.518592259 and -1.518592259.

Subproblem 1

n + -0.1428571429 = 1.518592259 Simplifying n + -0.1428571429 = 1.518592259 Reorder the terms: -0.1428571429 + n = 1.518592259 Solving -0.1428571429 + n = 1.518592259 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + n = 1.518592259 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + n = 1.518592259 + 0.1428571429 n = 1.518592259 + 0.1428571429 Combine like terms: 1.518592259 + 0.1428571429 = 1.6614494019 n = 1.6614494019 Simplifying n = 1.6614494019

Subproblem 2

n + -0.1428571429 = -1.518592259 Simplifying n + -0.1428571429 = -1.518592259 Reorder the terms: -0.1428571429 + n = -1.518592259 Solving -0.1428571429 + n = -1.518592259 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + n = -1.518592259 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + n = -1.518592259 + 0.1428571429 n = -1.518592259 + 0.1428571429 Combine like terms: -1.518592259 + 0.1428571429 = -1.3757351161 n = -1.3757351161 Simplifying n = -1.3757351161

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.6614494019, -1.3757351161}

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