4n^2=-6n-1

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Solution for 4n^2=-6n-1 equation:


Simplifying
4n2 = -6n + -1

Reorder the terms:
4n2 = -1 + -6n

Solving
4n2 = -1 + -6n

Solving for variable 'n'.

Reorder the terms:
1 + 6n + 4n2 = -1 + -6n + 1 + 6n

Reorder the terms:
1 + 6n + 4n2 = -1 + 1 + -6n + 6n

Combine like terms: -1 + 1 = 0
1 + 6n + 4n2 = 0 + -6n + 6n
1 + 6n + 4n2 = -6n + 6n

Combine like terms: -6n + 6n = 0
1 + 6n + 4n2 = 0

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

Divide each side by '4'.
0.25 + 1.5n + n2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 1.5n + -0.25 + n2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 1.5n + n2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 1.5n + n2 = 0 + -0.25
1.5n + n2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
1.5n + n2 = -0.25

The n term is 1.5n.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5n + 0.5625 + n2 = -0.25 + 0.5625

Reorder the terms:
0.5625 + 1.5n + n2 = -0.25 + 0.5625

Combine like terms: -0.25 + 0.5625 = 0.3125
0.5625 + 1.5n + n2 = 0.3125

Factor a perfect square on the left side:
(n + 0.75)(n + 0.75) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(n + 0.75) equal to 0.559016994 and -0.559016994.

Subproblem 1

n + 0.75 = 0.559016994 Simplifying n + 0.75 = 0.559016994 Reorder the terms: 0.75 + n = 0.559016994 Solving 0.75 + n = 0.559016994 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = 0.559016994 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = 0.559016994 + -0.75 n = 0.559016994 + -0.75 Combine like terms: 0.559016994 + -0.75 = -0.190983006 n = -0.190983006 Simplifying n = -0.190983006

Subproblem 2

n + 0.75 = -0.559016994 Simplifying n + 0.75 = -0.559016994 Reorder the terms: 0.75 + n = -0.559016994 Solving 0.75 + n = -0.559016994 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = -0.559016994 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = -0.559016994 + -0.75 n = -0.559016994 + -0.75 Combine like terms: -0.559016994 + -0.75 = -1.309016994 n = -1.309016994 Simplifying n = -1.309016994

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.190983006, -1.309016994}

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