8n(4n-5)=7

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Solution for 8n(4n-5)=7 equation:


Simplifying
8n(4n + -5) = 7

Reorder the terms:
8n(-5 + 4n) = 7
(-5 * 8n + 4n * 8n) = 7
(-40n + 32n2) = 7

Solving
-40n + 32n2 = 7

Solving for variable 'n'.

Reorder the terms:
-7 + -40n + 32n2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + -40n + 32n2 = 0

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

Divide each side by '32'.
-0.21875 + -1.25n + n2 = 0

Move the constant term to the right:

Add '0.21875' to each side of the equation.
-0.21875 + -1.25n + 0.21875 + n2 = 0 + 0.21875

Reorder the terms:
-0.21875 + 0.21875 + -1.25n + n2 = 0 + 0.21875

Combine like terms: -0.21875 + 0.21875 = 0.00000
0.00000 + -1.25n + n2 = 0 + 0.21875
-1.25n + n2 = 0 + 0.21875

Combine like terms: 0 + 0.21875 = 0.21875
-1.25n + n2 = 0.21875

The n term is -1.25n.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25n + 0.390625 + n2 = 0.21875 + 0.390625

Reorder the terms:
0.390625 + -1.25n + n2 = 0.21875 + 0.390625

Combine like terms: 0.21875 + 0.390625 = 0.609375
0.390625 + -1.25n + n2 = 0.609375

Factor a perfect square on the left side:
(n + -0.625)(n + -0.625) = 0.609375

Calculate the square root of the right side: 0.78062475

Break this problem into two subproblems by setting 
(n + -0.625) equal to 0.78062475 and -0.78062475.

Subproblem 1

n + -0.625 = 0.78062475 Simplifying n + -0.625 = 0.78062475 Reorder the terms: -0.625 + n = 0.78062475 Solving -0.625 + n = 0.78062475 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + n = 0.78062475 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + n = 0.78062475 + 0.625 n = 0.78062475 + 0.625 Combine like terms: 0.78062475 + 0.625 = 1.40562475 n = 1.40562475 Simplifying n = 1.40562475

Subproblem 2

n + -0.625 = -0.78062475 Simplifying n + -0.625 = -0.78062475 Reorder the terms: -0.625 + n = -0.78062475 Solving -0.625 + n = -0.78062475 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + n = -0.78062475 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + n = -0.78062475 + 0.625 n = -0.78062475 + 0.625 Combine like terms: -0.78062475 + 0.625 = -0.15562475 n = -0.15562475 Simplifying n = -0.15562475

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.40562475, -0.15562475}

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