7n^2=-21n+16

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Solution for 7n^2=-21n+16 equation:


Simplifying
7n2 = -21n + 16

Reorder the terms:
7n2 = 16 + -21n

Solving
7n2 = 16 + -21n

Solving for variable 'n'.

Reorder the terms:
-16 + 21n + 7n2 = 16 + -21n + -16 + 21n

Reorder the terms:
-16 + 21n + 7n2 = 16 + -16 + -21n + 21n

Combine like terms: 16 + -16 = 0
-16 + 21n + 7n2 = 0 + -21n + 21n
-16 + 21n + 7n2 = -21n + 21n

Combine like terms: -21n + 21n = 0
-16 + 21n + 7n2 = 0

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

Divide each side by '7'.
-2.285714286 + 3n + n2 = 0

Move the constant term to the right:

Add '2.285714286' to each side of the equation.
-2.285714286 + 3n + 2.285714286 + n2 = 0 + 2.285714286

Reorder the terms:
-2.285714286 + 2.285714286 + 3n + n2 = 0 + 2.285714286

Combine like terms: -2.285714286 + 2.285714286 = 0.000000000
0.000000000 + 3n + n2 = 0 + 2.285714286
3n + n2 = 0 + 2.285714286

Combine like terms: 0 + 2.285714286 = 2.285714286
3n + n2 = 2.285714286

The n term is 3n.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3n + 2.25 + n2 = 2.285714286 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 2.285714286 + 2.25

Combine like terms: 2.285714286 + 2.25 = 4.535714286
2.25 + 3n + n2 = 4.535714286

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 4.535714286

Calculate the square root of the right side: 2.129721645

Break this problem into two subproblems by setting 
(n + 1.5) equal to 2.129721645 and -2.129721645.

Subproblem 1

n + 1.5 = 2.129721645 Simplifying n + 1.5 = 2.129721645 Reorder the terms: 1.5 + n = 2.129721645 Solving 1.5 + n = 2.129721645 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 2.129721645 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 2.129721645 + -1.5 n = 2.129721645 + -1.5 Combine like terms: 2.129721645 + -1.5 = 0.629721645 n = 0.629721645 Simplifying n = 0.629721645

Subproblem 2

n + 1.5 = -2.129721645 Simplifying n + 1.5 = -2.129721645 Reorder the terms: 1.5 + n = -2.129721645 Solving 1.5 + n = -2.129721645 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -2.129721645 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -2.129721645 + -1.5 n = -2.129721645 + -1.5 Combine like terms: -2.129721645 + -1.5 = -3.629721645 n = -3.629721645 Simplifying n = -3.629721645

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.629721645, -3.629721645}

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