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Simplifying -9v2 + 35v + -30 = 1 + -1v Reorder the terms: -30 + 35v + -9v2 = 1 + -1v Solving -30 + 35v + -9v2 = 1 + -1v Solving for variable 'v'. Reorder the terms: -30 + -1 + 35v + v + -9v2 = 1 + -1v + -1 + v Combine like terms: -30 + -1 = -31 -31 + 35v + v + -9v2 = 1 + -1v + -1 + v Combine like terms: 35v + v = 36v -31 + 36v + -9v2 = 1 + -1v + -1 + v Reorder the terms: -31 + 36v + -9v2 = 1 + -1 + -1v + v Combine like terms: 1 + -1 = 0 -31 + 36v + -9v2 = 0 + -1v + v -31 + 36v + -9v2 = -1v + v Combine like terms: -1v + v = 0 -31 + 36v + -9v2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. 3.444444444 + -4v + v2 = 0 Move the constant term to the right: Add '-3.444444444' to each side of the equation. 3.444444444 + -4v + -3.444444444 + v2 = 0 + -3.444444444 Reorder the terms: 3.444444444 + -3.444444444 + -4v + v2 = 0 + -3.444444444 Combine like terms: 3.444444444 + -3.444444444 = 0.000000000 0.000000000 + -4v + v2 = 0 + -3.444444444 -4v + v2 = 0 + -3.444444444 Combine like terms: 0 + -3.444444444 = -3.444444444 -4v + v2 = -3.444444444 The v term is -4v. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4v + 4 + v2 = -3.444444444 + 4 Reorder the terms: 4 + -4v + v2 = -3.444444444 + 4 Combine like terms: -3.444444444 + 4 = 0.555555556 4 + -4v + v2 = 0.555555556 Factor a perfect square on the left side: (v + -2)(v + -2) = 0.555555556 Calculate the square root of the right side: 0.745355993 Break this problem into two subproblems by setting (v + -2) equal to 0.745355993 and -0.745355993.Subproblem 1
v + -2 = 0.745355993 Simplifying v + -2 = 0.745355993 Reorder the terms: -2 + v = 0.745355993 Solving -2 + v = 0.745355993 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + v = 0.745355993 + 2 Combine like terms: -2 + 2 = 0 0 + v = 0.745355993 + 2 v = 0.745355993 + 2 Combine like terms: 0.745355993 + 2 = 2.745355993 v = 2.745355993 Simplifying v = 2.745355993Subproblem 2
v + -2 = -0.745355993 Simplifying v + -2 = -0.745355993 Reorder the terms: -2 + v = -0.745355993 Solving -2 + v = -0.745355993 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + v = -0.745355993 + 2 Combine like terms: -2 + 2 = 0 0 + v = -0.745355993 + 2 v = -0.745355993 + 2 Combine like terms: -0.745355993 + 2 = 1.254644007 v = 1.254644007 Simplifying v = 1.254644007Solution
The solution to the problem is based on the solutions from the subproblems. v = {2.745355993, 1.254644007}
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