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Simplifying 2v2 + 9v + -7 + 10v2 = 4v + 10 + v + -8 Reorder the terms: -7 + 9v + 2v2 + 10v2 = 4v + 10 + v + -8 Combine like terms: 2v2 + 10v2 = 12v2 -7 + 9v + 12v2 = 4v + 10 + v + -8 Reorder the terms: -7 + 9v + 12v2 = 10 + -8 + 4v + v Combine like terms: 10 + -8 = 2 -7 + 9v + 12v2 = 2 + 4v + v Combine like terms: 4v + v = 5v -7 + 9v + 12v2 = 2 + 5v Solving -7 + 9v + 12v2 = 2 + 5v Solving for variable 'v'. Reorder the terms: -7 + -2 + 9v + -5v + 12v2 = 2 + 5v + -2 + -5v Combine like terms: -7 + -2 = -9 -9 + 9v + -5v + 12v2 = 2 + 5v + -2 + -5v Combine like terms: 9v + -5v = 4v -9 + 4v + 12v2 = 2 + 5v + -2 + -5v Reorder the terms: -9 + 4v + 12v2 = 2 + -2 + 5v + -5v Combine like terms: 2 + -2 = 0 -9 + 4v + 12v2 = 0 + 5v + -5v -9 + 4v + 12v2 = 5v + -5v Combine like terms: 5v + -5v = 0 -9 + 4v + 12v2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.75 + 0.3333333333v + v2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 0.3333333333v + 0.75 + v2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 0.3333333333v + v2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 0.3333333333v + v2 = 0 + 0.75 0.3333333333v + v2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 0.3333333333v + v2 = 0.75 The v term is 0.3333333333v. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333v + 0.02777777779 + v2 = 0.75 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333v + v2 = 0.75 + 0.02777777779 Combine like terms: 0.75 + 0.02777777779 = 0.77777777779 0.02777777779 + 0.3333333333v + v2 = 0.77777777779 Factor a perfect square on the left side: (v + 0.1666666667)(v + 0.1666666667) = 0.77777777779 Calculate the square root of the right side: 0.881917104 Break this problem into two subproblems by setting (v + 0.1666666667) equal to 0.881917104 and -0.881917104.Subproblem 1
v + 0.1666666667 = 0.881917104 Simplifying v + 0.1666666667 = 0.881917104 Reorder the terms: 0.1666666667 + v = 0.881917104 Solving 0.1666666667 + v = 0.881917104 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + v = 0.881917104 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + v = 0.881917104 + -0.1666666667 v = 0.881917104 + -0.1666666667 Combine like terms: 0.881917104 + -0.1666666667 = 0.7152504373 v = 0.7152504373 Simplifying v = 0.7152504373Subproblem 2
v + 0.1666666667 = -0.881917104 Simplifying v + 0.1666666667 = -0.881917104 Reorder the terms: 0.1666666667 + v = -0.881917104 Solving 0.1666666667 + v = -0.881917104 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + v = -0.881917104 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + v = -0.881917104 + -0.1666666667 v = -0.881917104 + -0.1666666667 Combine like terms: -0.881917104 + -0.1666666667 = -1.0485837707 v = -1.0485837707 Simplifying v = -1.0485837707Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.7152504373, -1.0485837707}
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