7n^2-6n-21=0

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Solution for 7n^2-6n-21=0 equation:


Simplifying
7n2 + -6n + -21 = 0

Reorder the terms:
-21 + -6n + 7n2 = 0

Solving
-21 + -6n + 7n2 = 0

Solving for variable 'n'.

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

Divide each side by '7'.
-3 + -0.8571428571n + n2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -0.8571428571n + 3 + n2 = 0 + 3

Reorder the terms:
-3 + 3 + -0.8571428571n + n2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -0.8571428571n + n2 = 0 + 3
-0.8571428571n + n2 = 0 + 3

Combine like terms: 0 + 3 = 3
-0.8571428571n + n2 = 3

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 = 3 + 0.1836734694

Reorder the terms:
0.1836734694 + -0.8571428571n + n2 = 3 + 0.1836734694

Combine like terms: 3 + 0.1836734694 = 3.1836734694
0.1836734694 + -0.8571428571n + n2 = 3.1836734694

Factor a perfect square on the left side:
(n + -0.4285714286)(n + -0.4285714286) = 3.1836734694

Calculate the square root of the right side: 1.784285142

Break this problem into two subproblems by setting 
(n + -0.4285714286) equal to 1.784285142 and -1.784285142.

Subproblem 1

n + -0.4285714286 = 1.784285142 Simplifying n + -0.4285714286 = 1.784285142 Reorder the terms: -0.4285714286 + n = 1.784285142 Solving -0.4285714286 + n = 1.784285142 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 = 1.784285142 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + n = 1.784285142 + 0.4285714286 n = 1.784285142 + 0.4285714286 Combine like terms: 1.784285142 + 0.4285714286 = 2.2128565706 n = 2.2128565706 Simplifying n = 2.2128565706

Subproblem 2

n + -0.4285714286 = -1.784285142 Simplifying n + -0.4285714286 = -1.784285142 Reorder the terms: -0.4285714286 + n = -1.784285142 Solving -0.4285714286 + n = -1.784285142 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 = -1.784285142 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + n = -1.784285142 + 0.4285714286 n = -1.784285142 + 0.4285714286 Combine like terms: -1.784285142 + 0.4285714286 = -1.3557137134 n = -1.3557137134 Simplifying n = -1.3557137134

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.2128565706, -1.3557137134}

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