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Simplifying (3v * 3v) + 14v + -5 = 0 Reorder the terms for easier multiplication: (3 * 3v * v) + 14v + -5 = 0 Multiply 3 * 3 (9v * v) + 14v + -5 = 0 Multiply v * v (9v2) + 14v + -5 = 0 Reorder the terms: -5 + 14v + (9v2) = 0 Solving -5 + 14v + (9v2) = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.5555555556 + 1.555555556v + v2 = 0 Move the constant term to the right: Add '0.5555555556' to each side of the equation. -0.5555555556 + 1.555555556v + 0.5555555556 + v2 = 0 + 0.5555555556 Reorder the terms: -0.5555555556 + 0.5555555556 + 1.555555556v + v2 = 0 + 0.5555555556 Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000 0.0000000000 + 1.555555556v + v2 = 0 + 0.5555555556 1.555555556v + v2 = 0 + 0.5555555556 Combine like terms: 0 + 0.5555555556 = 0.5555555556 1.555555556v + v2 = 0.5555555556 The v term is 1.555555556v. Take half its coefficient (0.777777778). Square it (0.6049382720) and add it to both sides. Add '0.6049382720' to each side of the equation. 1.555555556v + 0.6049382720 + v2 = 0.5555555556 + 0.6049382720 Reorder the terms: 0.6049382720 + 1.555555556v + v2 = 0.5555555556 + 0.6049382720 Combine like terms: 0.5555555556 + 0.6049382720 = 1.1604938276 0.6049382720 + 1.555555556v + v2 = 1.1604938276 Factor a perfect square on the left side: ((v) + 0.777777778)((v) + 0.777777778) = 1.1604938276 Calculate the square root of the right side: 1.077262191 Break this problem into two subproblems by setting ((v) + 0.777777778) equal to 1.077262191 and -1.077262191.Subproblem 1
(v) + 0.777777778 = 1.077262191 Simplifying (v) + 0.777777778 = 1.077262191 v + 0.777777778 = 1.077262191 Reorder the terms: 0.777777778 + v = 1.077262191 Solving 0.777777778 + v = 1.077262191 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.777777778' to each side of the equation. 0.777777778 + -0.777777778 + v = 1.077262191 + -0.777777778 Combine like terms: 0.777777778 + -0.777777778 = 0.000000000 0.000000000 + v = 1.077262191 + -0.777777778 v = 1.077262191 + -0.777777778 Combine like terms: 1.077262191 + -0.777777778 = 0.299484413 v = 0.299484413 Simplifying v = 0.299484413Subproblem 2
(v) + 0.777777778 = -1.077262191 Simplifying (v) + 0.777777778 = -1.077262191 v + 0.777777778 = -1.077262191 Reorder the terms: 0.777777778 + v = -1.077262191 Solving 0.777777778 + v = -1.077262191 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.777777778' to each side of the equation. 0.777777778 + -0.777777778 + v = -1.077262191 + -0.777777778 Combine like terms: 0.777777778 + -0.777777778 = 0.000000000 0.000000000 + v = -1.077262191 + -0.777777778 v = -1.077262191 + -0.777777778 Combine like terms: -1.077262191 + -0.777777778 = -1.855039969 v = -1.855039969 Simplifying v = -1.855039969Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.299484413, -1.855039969}
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