315=0.03*v^2+v

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Solution for 315=0.03*v^2+v equation:


Simplifying
315 = 0.03v2 + v

Reorder the terms:
315 = v + 0.03v2

Solving
315 = v + 0.03v2

Solving for variable 'v'.

Reorder the terms:
315 + -1v + -0.03v2 = v + -1v + 0.03v2 + -0.03v2

Combine like terms: v + -1v = 0
315 + -1v + -0.03v2 = 0 + 0.03v2 + -0.03v2
315 + -1v + -0.03v2 = 0.03v2 + -0.03v2

Combine like terms: 0.03v2 + -0.03v2 = 0.00
315 + -1v + -0.03v2 = 0.00

Begin completing the square.  Divide all terms by
-0.03 the coefficient of the squared term: 

Divide each side by '-0.03'.
-10500 + 33.33333333v + v2 = 0

Move the constant term to the right:

Add '10500' to each side of the equation.
-10500 + 33.33333333v + 10500 + v2 = 0 + 10500

Reorder the terms:
-10500 + 10500 + 33.33333333v + v2 = 0 + 10500

Combine like terms: -10500 + 10500 = 0
0 + 33.33333333v + v2 = 0 + 10500
33.33333333v + v2 = 0 + 10500

Combine like terms: 0 + 10500 = 10500
33.33333333v + v2 = 10500

The v term is 33.33333333v.  Take half its coefficient (16.66666667).
Square it (277.7777779) and add it to both sides.

Add '277.7777779' to each side of the equation.
33.33333333v + 277.7777779 + v2 = 10500 + 277.7777779

Reorder the terms:
277.7777779 + 33.33333333v + v2 = 10500 + 277.7777779

Combine like terms: 10500 + 277.7777779 = 10777.7777779
277.7777779 + 33.33333333v + v2 = 10777.7777779

Factor a perfect square on the left side:
(v + 16.66666667)(v + 16.66666667) = 10777.7777779

Calculate the square root of the right side: 103.816076683

Break this problem into two subproblems by setting 
(v + 16.66666667) equal to 103.816076683 and -103.816076683.

Subproblem 1

v + 16.66666667 = 103.816076683 Simplifying v + 16.66666667 = 103.816076683 Reorder the terms: 16.66666667 + v = 103.816076683 Solving 16.66666667 + v = 103.816076683 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-16.66666667' to each side of the equation. 16.66666667 + -16.66666667 + v = 103.816076683 + -16.66666667 Combine like terms: 16.66666667 + -16.66666667 = 0.00000000 0.00000000 + v = 103.816076683 + -16.66666667 v = 103.816076683 + -16.66666667 Combine like terms: 103.816076683 + -16.66666667 = 87.149410013 v = 87.149410013 Simplifying v = 87.149410013

Subproblem 2

v + 16.66666667 = -103.816076683 Simplifying v + 16.66666667 = -103.816076683 Reorder the terms: 16.66666667 + v = -103.816076683 Solving 16.66666667 + v = -103.816076683 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-16.66666667' to each side of the equation. 16.66666667 + -16.66666667 + v = -103.816076683 + -16.66666667 Combine like terms: 16.66666667 + -16.66666667 = 0.00000000 0.00000000 + v = -103.816076683 + -16.66666667 v = -103.816076683 + -16.66666667 Combine like terms: -103.816076683 + -16.66666667 = -120.482743353 v = -120.482743353 Simplifying v = -120.482743353

Solution

The solution to the problem is based on the solutions from the subproblems. v = {87.149410013, -120.482743353}

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