-7=8n^2+7n-15

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Solution for -7=8n^2+7n-15 equation:


Simplifying
-7 = 8n2 + 7n + -15

Reorder the terms:
-7 = -15 + 7n + 8n2

Solving
-7 = -15 + 7n + 8n2

Solving for variable 'n'.

Combine like terms: -7 + 15 = 8
8 + -7n + -8n2 = -15 + 7n + 8n2 + 15 + -7n + -8n2

Reorder the terms:
8 + -7n + -8n2 = -15 + 15 + 7n + -7n + 8n2 + -8n2

Combine like terms: -15 + 15 = 0
8 + -7n + -8n2 = 0 + 7n + -7n + 8n2 + -8n2
8 + -7n + -8n2 = 7n + -7n + 8n2 + -8n2

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

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

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
-1 + 0.875n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.875n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.875n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.875n + n2 = 0 + 1
0.875n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.875n + n2 = 1

The n term is 0.875n.  Take half its coefficient (0.4375).
Square it (0.19140625) and add it to both sides.

Add '0.19140625' to each side of the equation.
0.875n + 0.19140625 + n2 = 1 + 0.19140625

Reorder the terms:
0.19140625 + 0.875n + n2 = 1 + 0.19140625

Combine like terms: 1 + 0.19140625 = 1.19140625
0.19140625 + 0.875n + n2 = 1.19140625

Factor a perfect square on the left side:
(n + 0.4375)(n + 0.4375) = 1.19140625

Calculate the square root of the right side: 1.091515575

Break this problem into two subproblems by setting 
(n + 0.4375) equal to 1.091515575 and -1.091515575.

Subproblem 1

n + 0.4375 = 1.091515575 Simplifying n + 0.4375 = 1.091515575 Reorder the terms: 0.4375 + n = 1.091515575 Solving 0.4375 + n = 1.091515575 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + n = 1.091515575 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + n = 1.091515575 + -0.4375 n = 1.091515575 + -0.4375 Combine like terms: 1.091515575 + -0.4375 = 0.654015575 n = 0.654015575 Simplifying n = 0.654015575

Subproblem 2

n + 0.4375 = -1.091515575 Simplifying n + 0.4375 = -1.091515575 Reorder the terms: 0.4375 + n = -1.091515575 Solving 0.4375 + n = -1.091515575 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + n = -1.091515575 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + n = -1.091515575 + -0.4375 n = -1.091515575 + -0.4375 Combine like terms: -1.091515575 + -0.4375 = -1.529015575 n = -1.529015575 Simplifying n = -1.529015575

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.654015575, -1.529015575}

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