7n^2-5n-14=21

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Solution for 7n^2-5n-14=21 equation:


Simplifying
7n2 + -5n + -14 = 21

Reorder the terms:
-14 + -5n + 7n2 = 21

Solving
-14 + -5n + 7n2 = 21

Solving for variable 'n'.

Reorder the terms:
-14 + -21 + -5n + 7n2 = 21 + -21

Combine like terms: -14 + -21 = -35
-35 + -5n + 7n2 = 21 + -21

Combine like terms: 21 + -21 = 0
-35 + -5n + 7n2 = 0

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

Divide each side by '7'.
-5 + -0.7142857143n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -0.7142857143n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -0.7142857143n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -0.7142857143n + n2 = 0 + 5
-0.7142857143n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-0.7142857143n + n2 = 5

The n term is -0.7142857143n.  Take half its coefficient (-0.3571428572).
Square it (0.1275510204) and add it to both sides.

Add '0.1275510204' to each side of the equation.
-0.7142857143n + 0.1275510204 + n2 = 5 + 0.1275510204

Reorder the terms:
0.1275510204 + -0.7142857143n + n2 = 5 + 0.1275510204

Combine like terms: 5 + 0.1275510204 = 5.1275510204
0.1275510204 + -0.7142857143n + n2 = 5.1275510204

Factor a perfect square on the left side:
(n + -0.3571428572)(n + -0.3571428572) = 5.1275510204

Calculate the square root of the right side: 2.264409641

Break this problem into two subproblems by setting 
(n + -0.3571428572) equal to 2.264409641 and -2.264409641.

Subproblem 1

n + -0.3571428572 = 2.264409641 Simplifying n + -0.3571428572 = 2.264409641 Reorder the terms: -0.3571428572 + n = 2.264409641 Solving -0.3571428572 + n = 2.264409641 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + n = 2.264409641 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + n = 2.264409641 + 0.3571428572 n = 2.264409641 + 0.3571428572 Combine like terms: 2.264409641 + 0.3571428572 = 2.6215524982 n = 2.6215524982 Simplifying n = 2.6215524982

Subproblem 2

n + -0.3571428572 = -2.264409641 Simplifying n + -0.3571428572 = -2.264409641 Reorder the terms: -0.3571428572 + n = -2.264409641 Solving -0.3571428572 + n = -2.264409641 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + n = -2.264409641 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + n = -2.264409641 + 0.3571428572 n = -2.264409641 + 0.3571428572 Combine like terms: -2.264409641 + 0.3571428572 = -1.9072667838 n = -1.9072667838 Simplifying n = -1.9072667838

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.6215524982, -1.9072667838}

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