4n^2+3n-8=0

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Solution for 4n^2+3n-8=0 equation:


Simplifying
4n2 + 3n + -8 = 0

Reorder the terms:
-8 + 3n + 4n2 = 0

Solving
-8 + 3n + 4n2 = 0

Solving for variable 'n'.

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

Divide each side by '4'.
-2 + 0.75n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 0.75n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + 0.75n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 0.75n + n2 = 0 + 2
0.75n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
0.75n + n2 = 2

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 = 2 + 0.140625

Reorder the terms:
0.140625 + 0.75n + n2 = 2 + 0.140625

Combine like terms: 2 + 0.140625 = 2.140625
0.140625 + 0.75n + n2 = 2.140625

Factor a perfect square on the left side:
(n + 0.375)(n + 0.375) = 2.140625

Calculate the square root of the right side: 1.463087489

Break this problem into two subproblems by setting 
(n + 0.375) equal to 1.463087489 and -1.463087489.

Subproblem 1

n + 0.375 = 1.463087489 Simplifying n + 0.375 = 1.463087489 Reorder the terms: 0.375 + n = 1.463087489 Solving 0.375 + n = 1.463087489 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 = 1.463087489 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + n = 1.463087489 + -0.375 n = 1.463087489 + -0.375 Combine like terms: 1.463087489 + -0.375 = 1.088087489 n = 1.088087489 Simplifying n = 1.088087489

Subproblem 2

n + 0.375 = -1.463087489 Simplifying n + 0.375 = -1.463087489 Reorder the terms: 0.375 + n = -1.463087489 Solving 0.375 + n = -1.463087489 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 = -1.463087489 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + n = -1.463087489 + -0.375 n = -1.463087489 + -0.375 Combine like terms: -1.463087489 + -0.375 = -1.838087489 n = -1.838087489 Simplifying n = -1.838087489

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.088087489, -1.838087489}

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