8v^2+10v=9

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Solution for 8v^2+10v=9 equation:


Simplifying
8v2 + 10v = 9

Reorder the terms:
10v + 8v2 = 9

Solving
10v + 8v2 = 9

Solving for variable 'v'.

Reorder the terms:
-9 + 10v + 8v2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 10v + 8v2 = 0

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

Divide each side by '8'.
-1.125 + 1.25v + v2 = 0

Move the constant term to the right:

Add '1.125' to each side of the equation.
-1.125 + 1.25v + 1.125 + v2 = 0 + 1.125

Reorder the terms:
-1.125 + 1.125 + 1.25v + v2 = 0 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + 1.25v + v2 = 0 + 1.125
1.25v + v2 = 0 + 1.125

Combine like terms: 0 + 1.125 = 1.125
1.25v + v2 = 1.125

The v term is 1.25v.  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.
1.25v + 0.390625 + v2 = 1.125 + 0.390625

Reorder the terms:
0.390625 + 1.25v + v2 = 1.125 + 0.390625

Combine like terms: 1.125 + 0.390625 = 1.515625
0.390625 + 1.25v + v2 = 1.515625

Factor a perfect square on the left side:
(v + 0.625)(v + 0.625) = 1.515625

Calculate the square root of the right side: 1.231107225

Break this problem into two subproblems by setting 
(v + 0.625) equal to 1.231107225 and -1.231107225.

Subproblem 1

v + 0.625 = 1.231107225 Simplifying v + 0.625 = 1.231107225 Reorder the terms: 0.625 + v = 1.231107225 Solving 0.625 + v = 1.231107225 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + v = 1.231107225 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + v = 1.231107225 + -0.625 v = 1.231107225 + -0.625 Combine like terms: 1.231107225 + -0.625 = 0.606107225 v = 0.606107225 Simplifying v = 0.606107225

Subproblem 2

v + 0.625 = -1.231107225 Simplifying v + 0.625 = -1.231107225 Reorder the terms: 0.625 + v = -1.231107225 Solving 0.625 + v = -1.231107225 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + v = -1.231107225 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + v = -1.231107225 + -0.625 v = -1.231107225 + -0.625 Combine like terms: -1.231107225 + -0.625 = -1.856107225 v = -1.856107225 Simplifying v = -1.856107225

Solution

The solution to the problem is based on the solutions from the subproblems. v = {0.606107225, -1.856107225}

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