7n^2-3n-14=21

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Solution for 7n^2-3n-14=21 equation:


Simplifying
7n2 + -3n + -14 = 21

Reorder the terms:
-14 + -3n + 7n2 = 21

Solving
-14 + -3n + 7n2 = 21

Solving for variable 'n'.

Reorder the terms:
-14 + -21 + -3n + 7n2 = 21 + -21

Combine like terms: -14 + -21 = -35
-35 + -3n + 7n2 = 21 + -21

Combine like terms: 21 + -21 = 0
-35 + -3n + 7n2 = 0

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

Divide each side by '7'.
-5 + -0.4285714286n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -0.4285714286n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -0.4285714286n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -0.4285714286n + n2 = 0 + 5
-0.4285714286n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-0.4285714286n + n2 = 5

The n term is -0.4285714286n.  Take half its coefficient (-0.2142857143).
Square it (0.04591836735) and add it to both sides.

Add '0.04591836735' to each side of the equation.
-0.4285714286n + 0.04591836735 + n2 = 5 + 0.04591836735

Reorder the terms:
0.04591836735 + -0.4285714286n + n2 = 5 + 0.04591836735

Combine like terms: 5 + 0.04591836735 = 5.04591836735
0.04591836735 + -0.4285714286n + n2 = 5.04591836735

Factor a perfect square on the left side:
(n + -0.2142857143)(n + -0.2142857143) = 5.04591836735

Calculate the square root of the right side: 2.246312171

Break this problem into two subproblems by setting 
(n + -0.2142857143) equal to 2.246312171 and -2.246312171.

Subproblem 1

n + -0.2142857143 = 2.246312171 Simplifying n + -0.2142857143 = 2.246312171 Reorder the terms: -0.2142857143 + n = 2.246312171 Solving -0.2142857143 + n = 2.246312171 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2142857143' to each side of the equation. -0.2142857143 + 0.2142857143 + n = 2.246312171 + 0.2142857143 Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000 0.0000000000 + n = 2.246312171 + 0.2142857143 n = 2.246312171 + 0.2142857143 Combine like terms: 2.246312171 + 0.2142857143 = 2.4605978853 n = 2.4605978853 Simplifying n = 2.4605978853

Subproblem 2

n + -0.2142857143 = -2.246312171 Simplifying n + -0.2142857143 = -2.246312171 Reorder the terms: -0.2142857143 + n = -2.246312171 Solving -0.2142857143 + n = -2.246312171 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2142857143' to each side of the equation. -0.2142857143 + 0.2142857143 + n = -2.246312171 + 0.2142857143 Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000 0.0000000000 + n = -2.246312171 + 0.2142857143 n = -2.246312171 + 0.2142857143 Combine like terms: -2.246312171 + 0.2142857143 = -2.0320264567 n = -2.0320264567 Simplifying n = -2.0320264567

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.4605978853, -2.0320264567}

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