4n^2-3n-8=12

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Solution for 4n^2-3n-8=12 equation:


Simplifying
4n2 + -3n + -8 = 12

Reorder the terms:
-8 + -3n + 4n2 = 12

Solving
-8 + -3n + 4n2 = 12

Solving for variable 'n'.

Reorder the terms:
-8 + -12 + -3n + 4n2 = 12 + -12

Combine like terms: -8 + -12 = -20
-20 + -3n + 4n2 = 12 + -12

Combine like terms: 12 + -12 = 0
-20 + -3n + 4n2 = 0

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

Divide each side by '4'.
-5 + -0.75n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -0.75n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -0.75n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -0.75n + n2 = 0 + 5
-0.75n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-0.75n + n2 = 5

The n term is -0.75n.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75n + 0.140625 + n2 = 5 + 0.140625

Reorder the terms:
0.140625 + -0.75n + n2 = 5 + 0.140625

Combine like terms: 5 + 0.140625 = 5.140625
0.140625 + -0.75n + n2 = 5.140625

Factor a perfect square on the left side:
(n + -0.375)(n + -0.375) = 5.140625

Calculate the square root of the right side: 2.267294643

Break this problem into two subproblems by setting 
(n + -0.375) equal to 2.267294643 and -2.267294643.

Subproblem 1

n + -0.375 = 2.267294643 Simplifying n + -0.375 = 2.267294643 Reorder the terms: -0.375 + n = 2.267294643 Solving -0.375 + n = 2.267294643 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + n = 2.267294643 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = 2.267294643 + 0.375 n = 2.267294643 + 0.375 Combine like terms: 2.267294643 + 0.375 = 2.642294643 n = 2.642294643 Simplifying n = 2.642294643

Subproblem 2

n + -0.375 = -2.267294643 Simplifying n + -0.375 = -2.267294643 Reorder the terms: -0.375 + n = -2.267294643 Solving -0.375 + n = -2.267294643 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + n = -2.267294643 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = -2.267294643 + 0.375 n = -2.267294643 + 0.375 Combine like terms: -2.267294643 + 0.375 = -1.892294643 n = -1.892294643 Simplifying n = -1.892294643

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.642294643, -1.892294643}

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