10n^2-7=-4n

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Solution for 10n^2-7=-4n equation:


Simplifying
10n2 + -7 = -4n

Reorder the terms:
-7 + 10n2 = -4n

Solving
-7 + 10n2 = -4n

Solving for variable 'n'.

Reorder the terms:
-7 + 4n + 10n2 = -4n + 4n

Combine like terms: -4n + 4n = 0
-7 + 4n + 10n2 = 0

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

Divide each side by '10'.
-0.7 + 0.4n + n2 = 0

Move the constant term to the right:

Add '0.7' to each side of the equation.
-0.7 + 0.4n + 0.7 + n2 = 0 + 0.7

Reorder the terms:
-0.7 + 0.7 + 0.4n + n2 = 0 + 0.7

Combine like terms: -0.7 + 0.7 = 0.0
0.0 + 0.4n + n2 = 0 + 0.7
0.4n + n2 = 0 + 0.7

Combine like terms: 0 + 0.7 = 0.7
0.4n + n2 = 0.7

The n term is 0.4n.  Take half its coefficient (0.2).
Square it (0.04) and add it to both sides.

Add '0.04' to each side of the equation.
0.4n + 0.04 + n2 = 0.7 + 0.04

Reorder the terms:
0.04 + 0.4n + n2 = 0.7 + 0.04

Combine like terms: 0.7 + 0.04 = 0.74
0.04 + 0.4n + n2 = 0.74

Factor a perfect square on the left side:
(n + 0.2)(n + 0.2) = 0.74

Calculate the square root of the right side: 0.860232527

Break this problem into two subproblems by setting 
(n + 0.2) equal to 0.860232527 and -0.860232527.

Subproblem 1

n + 0.2 = 0.860232527 Simplifying n + 0.2 = 0.860232527 Reorder the terms: 0.2 + n = 0.860232527 Solving 0.2 + n = 0.860232527 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + n = 0.860232527 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + n = 0.860232527 + -0.2 n = 0.860232527 + -0.2 Combine like terms: 0.860232527 + -0.2 = 0.660232527 n = 0.660232527 Simplifying n = 0.660232527

Subproblem 2

n + 0.2 = -0.860232527 Simplifying n + 0.2 = -0.860232527 Reorder the terms: 0.2 + n = -0.860232527 Solving 0.2 + n = -0.860232527 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + n = -0.860232527 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + n = -0.860232527 + -0.2 n = -0.860232527 + -0.2 Combine like terms: -0.860232527 + -0.2 = -1.060232527 n = -1.060232527 Simplifying n = -1.060232527

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.660232527, -1.060232527}

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