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Simplifying 4v2 + -12v + -14 = -11 Reorder the terms: -14 + -12v + 4v2 = -11 Solving -14 + -12v + 4v2 = -11 Solving for variable 'v'. Reorder the terms: -14 + 11 + -12v + 4v2 = -11 + 11 Combine like terms: -14 + 11 = -3 -3 + -12v + 4v2 = -11 + 11 Combine like terms: -11 + 11 = 0 -3 + -12v + 4v2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -3v + v2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -3v + 0.75 + v2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -3v + v2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -3v + v2 = 0 + 0.75 -3v + v2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -3v + v2 = 0.75 The v term is -3v. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3v + 2.25 + v2 = 0.75 + 2.25 Reorder the terms: 2.25 + -3v + v2 = 0.75 + 2.25 Combine like terms: 0.75 + 2.25 = 3 2.25 + -3v + v2 = 3 Factor a perfect square on the left side: (v + -1.5)(v + -1.5) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (v + -1.5) equal to 1.732050808 and -1.732050808.Subproblem 1
v + -1.5 = 1.732050808 Simplifying v + -1.5 = 1.732050808 Reorder the terms: -1.5 + v = 1.732050808 Solving -1.5 + v = 1.732050808 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + v = 1.732050808 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + v = 1.732050808 + 1.5 v = 1.732050808 + 1.5 Combine like terms: 1.732050808 + 1.5 = 3.232050808 v = 3.232050808 Simplifying v = 3.232050808Subproblem 2
v + -1.5 = -1.732050808 Simplifying v + -1.5 = -1.732050808 Reorder the terms: -1.5 + v = -1.732050808 Solving -1.5 + v = -1.732050808 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + v = -1.732050808 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + v = -1.732050808 + 1.5 v = -1.732050808 + 1.5 Combine like terms: -1.732050808 + 1.5 = -0.232050808 v = -0.232050808 Simplifying v = -0.232050808Solution
The solution to the problem is based on the solutions from the subproblems. v = {3.232050808, -0.232050808}
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