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Simplifying 4v + -3v(v + -6) = 18 + -4v Reorder the terms: 4v + -3v(-6 + v) = 18 + -4v 4v + (-6 * -3v + v * -3v) = 18 + -4v 4v + (18v + -3v2) = 18 + -4v Combine like terms: 4v + 18v = 22v 22v + -3v2 = 18 + -4v Solving 22v + -3v2 = 18 + -4v Solving for variable 'v'. Reorder the terms: -18 + 22v + 4v + -3v2 = 18 + -4v + -18 + 4v Combine like terms: 22v + 4v = 26v -18 + 26v + -3v2 = 18 + -4v + -18 + 4v Reorder the terms: -18 + 26v + -3v2 = 18 + -18 + -4v + 4v Combine like terms: 18 + -18 = 0 -18 + 26v + -3v2 = 0 + -4v + 4v -18 + 26v + -3v2 = -4v + 4v Combine like terms: -4v + 4v = 0 -18 + 26v + -3v2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 6 + -8.666666667v + v2 = 0 Move the constant term to the right: Add '-6' to each side of the equation. 6 + -8.666666667v + -6 + v2 = 0 + -6 Reorder the terms: 6 + -6 + -8.666666667v + v2 = 0 + -6 Combine like terms: 6 + -6 = 0 0 + -8.666666667v + v2 = 0 + -6 -8.666666667v + v2 = 0 + -6 Combine like terms: 0 + -6 = -6 -8.666666667v + v2 = -6 The v term is -8.666666667v. Take half its coefficient (-4.333333334). Square it (18.77777778) and add it to both sides. Add '18.77777778' to each side of the equation. -8.666666667v + 18.77777778 + v2 = -6 + 18.77777778 Reorder the terms: 18.77777778 + -8.666666667v + v2 = -6 + 18.77777778 Combine like terms: -6 + 18.77777778 = 12.77777778 18.77777778 + -8.666666667v + v2 = 12.77777778 Factor a perfect square on the left side: (v + -4.333333334)(v + -4.333333334) = 12.77777778 Calculate the square root of the right side: 3.574601765 Break this problem into two subproblems by setting (v + -4.333333334) equal to 3.574601765 and -3.574601765.Subproblem 1
v + -4.333333334 = 3.574601765 Simplifying v + -4.333333334 = 3.574601765 Reorder the terms: -4.333333334 + v = 3.574601765 Solving -4.333333334 + v = 3.574601765 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '4.333333334' to each side of the equation. -4.333333334 + 4.333333334 + v = 3.574601765 + 4.333333334 Combine like terms: -4.333333334 + 4.333333334 = 0.000000000 0.000000000 + v = 3.574601765 + 4.333333334 v = 3.574601765 + 4.333333334 Combine like terms: 3.574601765 + 4.333333334 = 7.907935099 v = 7.907935099 Simplifying v = 7.907935099Subproblem 2
v + -4.333333334 = -3.574601765 Simplifying v + -4.333333334 = -3.574601765 Reorder the terms: -4.333333334 + v = -3.574601765 Solving -4.333333334 + v = -3.574601765 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '4.333333334' to each side of the equation. -4.333333334 + 4.333333334 + v = -3.574601765 + 4.333333334 Combine like terms: -4.333333334 + 4.333333334 = 0.000000000 0.000000000 + v = -3.574601765 + 4.333333334 v = -3.574601765 + 4.333333334 Combine like terms: -3.574601765 + 4.333333334 = 0.758731569 v = 0.758731569 Simplifying v = 0.758731569Solution
The solution to the problem is based on the solutions from the subproblems. v = {7.907935099, 0.758731569}
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