4n^2-8n=7

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Solution for 4n^2-8n=7 equation:


Simplifying
4n2 + -8n = 7

Reorder the terms:
-8n + 4n2 = 7

Solving
-8n + 4n2 = 7

Solving for variable 'n'.

Reorder the terms:
-7 + -8n + 4n2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + -8n + 4n2 = 0

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

Divide each side by '4'.
-1.75 + -2n + n2 = 0

Move the constant term to the right:

Add '1.75' to each side of the equation.
-1.75 + -2n + 1.75 + n2 = 0 + 1.75

Reorder the terms:
-1.75 + 1.75 + -2n + n2 = 0 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + -2n + n2 = 0 + 1.75
-2n + n2 = 0 + 1.75

Combine like terms: 0 + 1.75 = 1.75
-2n + n2 = 1.75

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 1.75 + 1

Reorder the terms:
1 + -2n + n2 = 1.75 + 1

Combine like terms: 1.75 + 1 = 2.75
1 + -2n + n2 = 2.75

Factor a perfect square on the left side:
(n + -1)(n + -1) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(n + -1) equal to 1.658312395 and -1.658312395.

Subproblem 1

n + -1 = 1.658312395 Simplifying n + -1 = 1.658312395 Reorder the terms: -1 + n = 1.658312395 Solving -1 + n = 1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 1.658312395 + 1 Combine like terms: -1 + 1 = 0 0 + n = 1.658312395 + 1 n = 1.658312395 + 1 Combine like terms: 1.658312395 + 1 = 2.658312395 n = 2.658312395 Simplifying n = 2.658312395

Subproblem 2

n + -1 = -1.658312395 Simplifying n + -1 = -1.658312395 Reorder the terms: -1 + n = -1.658312395 Solving -1 + n = -1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -1.658312395 + 1 Combine like terms: -1 + 1 = 0 0 + n = -1.658312395 + 1 n = -1.658312395 + 1 Combine like terms: -1.658312395 + 1 = -0.658312395 n = -0.658312395 Simplifying n = -0.658312395

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.658312395, -0.658312395}

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