v*v+7*7=2v*2v+17v+45

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Solution for v*v+7*7=2v*2v+17v+45 equation: 


Simplifying
v * v + 7 * 7 = 2v * 2v + 17v + 45

Multiply v * v
v2 + 7 * 7 = 2v * 2v + 17v + 45

Multiply 7 * 7
v2 + 49 = 2v * 2v + 17v + 45

Reorder the terms:
49 + v2 = 2v * 2v + 17v + 45

Reorder the terms for easier multiplication:
49 + v2 = 2 * 2v * v + 17v + 45

Multiply 2 * 2
49 + v2 = 4v * v + 17v + 45

Multiply v * v
49 + v2 = 4v2 + 17v + 45

Reorder the terms:
49 + v2 = 45 + 17v + 4v2

Solving
49 + v2 = 45 + 17v + 4v2

Solving for variable 'v'.

Reorder the terms:
49 + -45 + -17v + v2 + -4v2 = 45 + 17v + 4v2 + -45 + -17v + -4v2

Combine like terms: 49 + -45 = 4
4 + -17v + v2 + -4v2 = 45 + 17v + 4v2 + -45 + -17v + -4v2

Combine like terms: v2 + -4v2 = -3v2
4 + -17v + -3v2 = 45 + 17v + 4v2 + -45 + -17v + -4v2

Reorder the terms:
4 + -17v + -3v2 = 45 + -45 + 17v + -17v + 4v2 + -4v2

Combine like terms: 45 + -45 = 0
4 + -17v + -3v2 = 0 + 17v + -17v + 4v2 + -4v2
4 + -17v + -3v2 = 17v + -17v + 4v2 + -4v2

Combine like terms: 17v + -17v = 0
4 + -17v + -3v2 = 0 + 4v2 + -4v2
4 + -17v + -3v2 = 4v2 + -4v2

Combine like terms: 4v2 + -4v2 = 0
4 + -17v + -3v2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-1.333333333 + 5.666666667v + v2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + 5.666666667v + 1.333333333 + v2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + 5.666666667v + v2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + 5.666666667v + v2 = 0 + 1.333333333
5.666666667v + v2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
5.666666667v + v2 = 1.333333333

The v term is 5.666666667v.  Take half its coefficient (2.833333334).
Square it (8.027777782) and add it to both sides.

Add '8.027777782' to each side of the equation.
5.666666667v + 8.027777782 + v2 = 1.333333333 + 8.027777782

Reorder the terms:
8.027777782 + 5.666666667v + v2 = 1.333333333 + 8.027777782

Combine like terms: 1.333333333 + 8.027777782 = 9.361111115
8.027777782 + 5.666666667v + v2 = 9.361111115

Factor a perfect square on the left side:
(v + 2.833333334)(v + 2.833333334) = 9.361111115

Calculate the square root of the right side: 3.059593292

Break this problem into two subproblems by setting 
(v + 2.833333334) equal to 3.059593292 and -3.059593292.


Subproblem 1
v + 2.833333334 = 3.059593292

Simplifying
v + 2.833333334 = 3.059593292

Reorder the terms:
2.833333334 + v = 3.059593292

Solving
2.833333334 + v = 3.059593292

Solving for variable 'v'.

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

Add '-2.833333334' to each side of the equation.
2.833333334 + -2.833333334 + v = 3.059593292 + -2.833333334

Combine like terms: 2.833333334 + -2.833333334 = 0.000000000
0.000000000 + v = 3.059593292 + -2.833333334
v = 3.059593292 + -2.833333334

Combine like terms: 3.059593292 + -2.833333334 = 0.226259958
v = 0.226259958

Simplifying
v = 0.226259958


Subproblem 2
v + 2.833333334 = -3.059593292

Simplifying
v + 2.833333334 = -3.059593292

Reorder the terms:
2.833333334 + v = -3.059593292

Solving
2.833333334 + v = -3.059593292

Solving for variable 'v'.

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

Add '-2.833333334' to each side of the equation.
2.833333334 + -2.833333334 + v = -3.059593292 + -2.833333334

Combine like terms: 2.833333334 + -2.833333334 = 0.000000000
0.000000000 + v = -3.059593292 + -2.833333334
v = -3.059593292 + -2.833333334

Combine like terms: -3.059593292 + -2.833333334 = -5.892926626
v = -5.892926626

Simplifying
v = -5.892926626


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {0.226259958, -5.892926626}

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