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Simplifying v2 + -16 = 2v2 + -14v + 21 Reorder the terms: -16 + v2 = 2v2 + -14v + 21 Reorder the terms: -16 + v2 = 21 + -14v + 2v2 Solving -16 + v2 = 21 + -14v + 2v2 Solving for variable 'v'. Reorder the terms: -16 + -21 + 14v + v2 + -2v2 = 21 + -14v + 2v2 + -21 + 14v + -2v2 Combine like terms: -16 + -21 = -37 -37 + 14v + v2 + -2v2 = 21 + -14v + 2v2 + -21 + 14v + -2v2 Combine like terms: v2 + -2v2 = -1v2 -37 + 14v + -1v2 = 21 + -14v + 2v2 + -21 + 14v + -2v2 Reorder the terms: -37 + 14v + -1v2 = 21 + -21 + -14v + 14v + 2v2 + -2v2 Combine like terms: 21 + -21 = 0 -37 + 14v + -1v2 = 0 + -14v + 14v + 2v2 + -2v2 -37 + 14v + -1v2 = -14v + 14v + 2v2 + -2v2 Combine like terms: -14v + 14v = 0 -37 + 14v + -1v2 = 0 + 2v2 + -2v2 -37 + 14v + -1v2 = 2v2 + -2v2 Combine like terms: 2v2 + -2v2 = 0 -37 + 14v + -1v2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 37 + -14v + v2 = 0 Move the constant term to the right: Add '-37' to each side of the equation. 37 + -14v + -37 + v2 = 0 + -37 Reorder the terms: 37 + -37 + -14v + v2 = 0 + -37 Combine like terms: 37 + -37 = 0 0 + -14v + v2 = 0 + -37 -14v + v2 = 0 + -37 Combine like terms: 0 + -37 = -37 -14v + v2 = -37 The v term is -14v. Take half its coefficient (-7). Square it (49) and add it to both sides. Add '49' to each side of the equation. -14v + 49 + v2 = -37 + 49 Reorder the terms: 49 + -14v + v2 = -37 + 49 Combine like terms: -37 + 49 = 12 49 + -14v + v2 = 12 Factor a perfect square on the left side: (v + -7)(v + -7) = 12 Calculate the square root of the right side: 3.464101615 Break this problem into two subproblems by setting (v + -7) equal to 3.464101615 and -3.464101615.Subproblem 1
v + -7 = 3.464101615 Simplifying v + -7 = 3.464101615 Reorder the terms: -7 + v = 3.464101615 Solving -7 + v = 3.464101615 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + v = 3.464101615 + 7 Combine like terms: -7 + 7 = 0 0 + v = 3.464101615 + 7 v = 3.464101615 + 7 Combine like terms: 3.464101615 + 7 = 10.464101615 v = 10.464101615 Simplifying v = 10.464101615Subproblem 2
v + -7 = -3.464101615 Simplifying v + -7 = -3.464101615 Reorder the terms: -7 + v = -3.464101615 Solving -7 + v = -3.464101615 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + v = -3.464101615 + 7 Combine like terms: -7 + 7 = 0 0 + v = -3.464101615 + 7 v = -3.464101615 + 7 Combine like terms: -3.464101615 + 7 = 3.535898385 v = 3.535898385 Simplifying v = 3.535898385Solution
The solution to the problem is based on the solutions from the subproblems. v = {10.464101615, 3.535898385}
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