10v^2+8v-1=0

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Solution for 10v^2+8v-1=0 equation:


Simplifying
10v2 + 8v + -1 = 0

Reorder the terms:
-1 + 8v + 10v2 = 0

Solving
-1 + 8v + 10v2 = 0

Solving for variable 'v'.

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

Divide each side by '10'.
-0.1 + 0.8v + v2 = 0

Move the constant term to the right:

Add '0.1' to each side of the equation.
-0.1 + 0.8v + 0.1 + v2 = 0 + 0.1

Reorder the terms:
-0.1 + 0.1 + 0.8v + v2 = 0 + 0.1

Combine like terms: -0.1 + 0.1 = 0.0
0.0 + 0.8v + v2 = 0 + 0.1
0.8v + v2 = 0 + 0.1

Combine like terms: 0 + 0.1 = 0.1
0.8v + v2 = 0.1

The v term is 0.8v.  Take half its coefficient (0.4).
Square it (0.16) and add it to both sides.

Add '0.16' to each side of the equation.
0.8v + 0.16 + v2 = 0.1 + 0.16

Reorder the terms:
0.16 + 0.8v + v2 = 0.1 + 0.16

Combine like terms: 0.1 + 0.16 = 0.26
0.16 + 0.8v + v2 = 0.26

Factor a perfect square on the left side:
(v + 0.4)(v + 0.4) = 0.26

Calculate the square root of the right side: 0.509901951

Break this problem into two subproblems by setting 
(v + 0.4) equal to 0.509901951 and -0.509901951.

Subproblem 1

v + 0.4 = 0.509901951 Simplifying v + 0.4 = 0.509901951 Reorder the terms: 0.4 + v = 0.509901951 Solving 0.4 + v = 0.509901951 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + v = 0.509901951 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + v = 0.509901951 + -0.4 v = 0.509901951 + -0.4 Combine like terms: 0.509901951 + -0.4 = 0.109901951 v = 0.109901951 Simplifying v = 0.109901951

Subproblem 2

v + 0.4 = -0.509901951 Simplifying v + 0.4 = -0.509901951 Reorder the terms: 0.4 + v = -0.509901951 Solving 0.4 + v = -0.509901951 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + v = -0.509901951 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + v = -0.509901951 + -0.4 v = -0.509901951 + -0.4 Combine like terms: -0.509901951 + -0.4 = -0.909901951 v = -0.909901951 Simplifying v = -0.909901951

Solution

The solution to the problem is based on the solutions from the subproblems. v = {0.109901951, -0.909901951}

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