1v^2-21v=-40

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Solution for 1v^2-21v=-40 equation:


Simplifying
1v2 + -21v = -40

Reorder the terms:
-21v + 1v2 = -40

Solving
-21v + 1v2 = -40

Solving for variable 'v'.

Reorder the terms:
40 + -21v + 1v2 = -40 + 40

Combine like terms: -40 + 40 = 0
40 + -21v + 1v2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-40' to each side of the equation.
40 + -21v + -40 + v2 = 0 + -40

Reorder the terms:
40 + -40 + -21v + v2 = 0 + -40

Combine like terms: 40 + -40 = 0
0 + -21v + v2 = 0 + -40
-21v + v2 = 0 + -40

Combine like terms: 0 + -40 = -40
-21v + v2 = -40

The v term is -21v.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21v + 110.25 + v2 = -40 + 110.25

Reorder the terms:
110.25 + -21v + v2 = -40 + 110.25

Combine like terms: -40 + 110.25 = 70.25
110.25 + -21v + v2 = 70.25

Factor a perfect square on the left side:
(v + -10.5)(v + -10.5) = 70.25

Calculate the square root of the right side: 8.381527307

Break this problem into two subproblems by setting 
(v + -10.5) equal to 8.381527307 and -8.381527307.

Subproblem 1

v + -10.5 = 8.381527307 Simplifying v + -10.5 = 8.381527307 Reorder the terms: -10.5 + v = 8.381527307 Solving -10.5 + v = 8.381527307 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + v = 8.381527307 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + v = 8.381527307 + 10.5 v = 8.381527307 + 10.5 Combine like terms: 8.381527307 + 10.5 = 18.881527307 v = 18.881527307 Simplifying v = 18.881527307

Subproblem 2

v + -10.5 = -8.381527307 Simplifying v + -10.5 = -8.381527307 Reorder the terms: -10.5 + v = -8.381527307 Solving -10.5 + v = -8.381527307 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + v = -8.381527307 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + v = -8.381527307 + 10.5 v = -8.381527307 + 10.5 Combine like terms: -8.381527307 + 10.5 = 2.118472693 v = 2.118472693 Simplifying v = 2.118472693

Solution

The solution to the problem is based on the solutions from the subproblems. v = {18.881527307, 2.118472693}

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