7v(7v)-42=-35v

Solution for 7v(7v)-42=-35v equation:

Simplifying
7v(7v) + -42 = -35v

Remove parenthesis around (7v)
7v * 7v + -42 = -35v

Reorder the terms for easier multiplication:
7 * 7v * v + -42 = -35v

Multiply 7 * 7
49v * v + -42 = -35v

Multiply v * v
49v2 + -42 = -35v

Reorder the terms:
-42 + 49v2 = -35v

Solving
-42 + 49v2 = -35v

Solving for variable 'v'.

Reorder the terms:
-42 + 35v + 49v2 = -35v + 35v

Combine like terms: -35v + 35v = 0
-42 + 35v + 49v2 = 0

Factor out the Greatest Common Factor (GCF), '7'.
7(-6 + 5v + 7v2) = 0

Ignore the factor 7.
Subproblem 1
Set the factor '(-6 + 5v + 7v2)' equal to zero and attempt to solve:

Simplifying
-6 + 5v + 7v2 = 0

Solving
-6 + 5v + 7v2 = 0

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

Divide each side by '7'.
-0.8571428571 + 0.7142857143v + v2 = 0

Move the constant term to the right:

Add '0.8571428571' to each side of the equation.
-0.8571428571 + 0.7142857143v + 0.8571428571 + v2 = 0 + 0.8571428571

Reorder the terms:
-0.8571428571 + 0.8571428571 + 0.7142857143v + v2 = 0 + 0.8571428571

Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000
0.0000000000 + 0.7142857143v + v2 = 0 + 0.8571428571
0.7142857143v + v2 = 0 + 0.8571428571

Combine like terms: 0 + 0.8571428571 = 0.8571428571
0.7142857143v + v2 = 0.8571428571

The v term is 0.7142857143v.  Take half its coefficient (0.3571428572).
Square it (0.1275510204) and add it to both sides.

Add '0.1275510204' to each side of the equation.
0.7142857143v + 0.1275510204 + v2 = 0.8571428571 + 0.1275510204

Reorder the terms:
0.1275510204 + 0.7142857143v + v2 = 0.8571428571 + 0.1275510204

Combine like terms: 0.8571428571 + 0.1275510204 = 0.9846938775
0.1275510204 + 0.7142857143v + v2 = 0.9846938775

Factor a perfect square on the left side:
(v + 0.3571428572)(v + 0.3571428572) = 0.9846938775

Calculate the square root of the right side: 0.992317428

Break this problem into two subproblems by setting 
(v + 0.3571428572) equal to 0.992317428 and -0.992317428.

Subproblem 1
v + 0.3571428572 = 0.992317428

Simplifying
v + 0.3571428572 = 0.992317428

Reorder the terms:
0.3571428572 + v = 0.992317428

Solving
0.3571428572 + v = 0.992317428

Solving for variable 'v'.

Move all terms containing v to the left, all other terms to the right.

Add '-0.3571428572' to each side of the equation.
0.3571428572 + -0.3571428572 + v = 0.992317428 + -0.3571428572

Combine like terms: 0.3571428572 + -0.3571428572 = 0.0000000000
0.0000000000 + v = 0.992317428 + -0.3571428572
v = 0.992317428 + -0.3571428572

Combine like terms: 0.992317428 + -0.3571428572 = 0.6351745708
v = 0.6351745708

Simplifying
v = 0.6351745708

Subproblem 2
v + 0.3571428572 = -0.992317428

Simplifying
v + 0.3571428572 = -0.992317428

Reorder the terms:
0.3571428572 + v = -0.992317428

Solving
0.3571428572 + v = -0.992317428

Solving for variable 'v'.

Move all terms containing v to the left, all other terms to the right.

Add '-0.3571428572' to each side of the equation.
0.3571428572 + -0.3571428572 + v = -0.992317428 + -0.3571428572

Combine like terms: 0.3571428572 + -0.3571428572 = 0.0000000000
0.0000000000 + v = -0.992317428 + -0.3571428572
v = -0.992317428 + -0.3571428572

Combine like terms: -0.992317428 + -0.3571428572 = -1.3494602852
v = -1.3494602852

Simplifying
v = -1.3494602852

Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {0.6351745708, -1.3494602852}
Solution
v = {0.6351745708, -1.3494602852}