7n^2-6n-14=21

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Solution for 7n^2-6n-14=21 equation:


Simplifying
7n2 + -6n + -14 = 21

Reorder the terms:
-14 + -6n + 7n2 = 21

Solving
-14 + -6n + 7n2 = 21

Solving for variable 'n'.

Reorder the terms:
-14 + -21 + -6n + 7n2 = 21 + -21

Combine like terms: -14 + -21 = -35
-35 + -6n + 7n2 = 21 + -21

Combine like terms: 21 + -21 = 0
-35 + -6n + 7n2 = 0

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

Divide each side by '7'.
-5 + -0.8571428571n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -0.8571428571n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -0.8571428571n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -0.8571428571n + n2 = 0 + 5
-0.8571428571n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-0.8571428571n + n2 = 5

The n term is -0.8571428571n.  Take half its coefficient (-0.4285714286).
Square it (0.1836734694) and add it to both sides.

Add '0.1836734694' to each side of the equation.
-0.8571428571n + 0.1836734694 + n2 = 5 + 0.1836734694

Reorder the terms:
0.1836734694 + -0.8571428571n + n2 = 5 + 0.1836734694

Combine like terms: 5 + 0.1836734694 = 5.1836734694
0.1836734694 + -0.8571428571n + n2 = 5.1836734694

Factor a perfect square on the left side:
(n + -0.4285714286)(n + -0.4285714286) = 5.1836734694

Calculate the square root of the right side: 2.276768207

Break this problem into two subproblems by setting 
(n + -0.4285714286) equal to 2.276768207 and -2.276768207.

Subproblem 1

n + -0.4285714286 = 2.276768207 Simplifying n + -0.4285714286 = 2.276768207 Reorder the terms: -0.4285714286 + n = 2.276768207 Solving -0.4285714286 + n = 2.276768207 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + n = 2.276768207 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + n = 2.276768207 + 0.4285714286 n = 2.276768207 + 0.4285714286 Combine like terms: 2.276768207 + 0.4285714286 = 2.7053396356 n = 2.7053396356 Simplifying n = 2.7053396356

Subproblem 2

n + -0.4285714286 = -2.276768207 Simplifying n + -0.4285714286 = -2.276768207 Reorder the terms: -0.4285714286 + n = -2.276768207 Solving -0.4285714286 + n = -2.276768207 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + n = -2.276768207 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + n = -2.276768207 + 0.4285714286 n = -2.276768207 + 0.4285714286 Combine like terms: -2.276768207 + 0.4285714286 = -1.8481967784 n = -1.8481967784 Simplifying n = -1.8481967784

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.7053396356, -1.8481967784}

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