7n^2-4n-14=21

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Solution for 7n^2-4n-14=21 equation:


Simplifying
7n2 + -4n + -14 = 21

Reorder the terms:
-14 + -4n + 7n2 = 21

Solving
-14 + -4n + 7n2 = 21

Solving for variable 'n'.

Reorder the terms:
-14 + -21 + -4n + 7n2 = 21 + -21

Combine like terms: -14 + -21 = -35
-35 + -4n + 7n2 = 21 + -21

Combine like terms: 21 + -21 = 0
-35 + -4n + 7n2 = 0

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

Divide each side by '7'.
-5 + -0.5714285714n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -0.5714285714n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -0.5714285714n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -0.5714285714n + n2 = 0 + 5
-0.5714285714n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-0.5714285714n + n2 = 5

The n term is -0.5714285714n.  Take half its coefficient (-0.2857142857).
Square it (0.08163265305) and add it to both sides.

Add '0.08163265305' to each side of the equation.
-0.5714285714n + 0.08163265305 + n2 = 5 + 0.08163265305

Reorder the terms:
0.08163265305 + -0.5714285714n + n2 = 5 + 0.08163265305

Combine like terms: 5 + 0.08163265305 = 5.08163265305
0.08163265305 + -0.5714285714n + n2 = 5.08163265305

Factor a perfect square on the left side:
(n + -0.2857142857)(n + -0.2857142857) = 5.08163265305

Calculate the square root of the right side: 2.254247691

Break this problem into two subproblems by setting 
(n + -0.2857142857) equal to 2.254247691 and -2.254247691.

Subproblem 1

n + -0.2857142857 = 2.254247691 Simplifying n + -0.2857142857 = 2.254247691 Reorder the terms: -0.2857142857 + n = 2.254247691 Solving -0.2857142857 + n = 2.254247691 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + n = 2.254247691 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + n = 2.254247691 + 0.2857142857 n = 2.254247691 + 0.2857142857 Combine like terms: 2.254247691 + 0.2857142857 = 2.5399619767 n = 2.5399619767 Simplifying n = 2.5399619767

Subproblem 2

n + -0.2857142857 = -2.254247691 Simplifying n + -0.2857142857 = -2.254247691 Reorder the terms: -0.2857142857 + n = -2.254247691 Solving -0.2857142857 + n = -2.254247691 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + n = -2.254247691 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + n = -2.254247691 + 0.2857142857 n = -2.254247691 + 0.2857142857 Combine like terms: -2.254247691 + 0.2857142857 = -1.9685334053 n = -1.9685334053 Simplifying n = -1.9685334053

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.5399619767, -1.9685334053}

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