6v^2-7=10v

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Solution for 6v^2-7=10v equation:


Simplifying
6v2 + -7 = 10v

Reorder the terms:
-7 + 6v2 = 10v

Solving
-7 + 6v2 = 10v

Solving for variable 'v'.

Reorder the terms:
-7 + -10v + 6v2 = 10v + -10v

Combine like terms: 10v + -10v = 0
-7 + -10v + 6v2 = 0

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

Divide each side by '6'.
-1.166666667 + -1.666666667v + v2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + -1.666666667v + 1.166666667 + v2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + -1.666666667v + v2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + -1.666666667v + v2 = 0 + 1.166666667
-1.666666667v + v2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
-1.666666667v + v2 = 1.166666667

The v term is -1.666666667v.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667v + 0.6944444447 + v2 = 1.166666667 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667v + v2 = 1.166666667 + 0.6944444447

Combine like terms: 1.166666667 + 0.6944444447 = 1.8611111117
0.6944444447 + -1.666666667v + v2 = 1.8611111117

Factor a perfect square on the left side:
(v + -0.8333333335)(v + -0.8333333335) = 1.8611111117

Calculate the square root of the right side: 1.364225462

Break this problem into two subproblems by setting 
(v + -0.8333333335) equal to 1.364225462 and -1.364225462.

Subproblem 1

v + -0.8333333335 = 1.364225462 Simplifying v + -0.8333333335 = 1.364225462 Reorder the terms: -0.8333333335 + v = 1.364225462 Solving -0.8333333335 + v = 1.364225462 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + v = 1.364225462 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + v = 1.364225462 + 0.8333333335 v = 1.364225462 + 0.8333333335 Combine like terms: 1.364225462 + 0.8333333335 = 2.1975587955 v = 2.1975587955 Simplifying v = 2.1975587955

Subproblem 2

v + -0.8333333335 = -1.364225462 Simplifying v + -0.8333333335 = -1.364225462 Reorder the terms: -0.8333333335 + v = -1.364225462 Solving -0.8333333335 + v = -1.364225462 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + v = -1.364225462 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + v = -1.364225462 + 0.8333333335 v = -1.364225462 + 0.8333333335 Combine like terms: -1.364225462 + 0.8333333335 = -0.5308921285 v = -0.5308921285 Simplifying v = -0.5308921285

Solution

The solution to the problem is based on the solutions from the subproblems. v = {2.1975587955, -0.5308921285}

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