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Simplifying -9(v + 3) * 2v + 7 = 5v + 12 Reorder the terms: -9(3 + v) * 2v + 7 = 5v + 12 Reorder the terms for easier multiplication: -9 * 2v(3 + v) + 7 = 5v + 12 Multiply -9 * 2 -18v(3 + v) + 7 = 5v + 12 (3 * -18v + v * -18v) + 7 = 5v + 12 (-54v + -18v2) + 7 = 5v + 12 Reorder the terms: 7 + -54v + -18v2 = 5v + 12 Reorder the terms: 7 + -54v + -18v2 = 12 + 5v Solving 7 + -54v + -18v2 = 12 + 5v Solving for variable 'v'. Reorder the terms: 7 + -12 + -54v + -5v + -18v2 = 12 + 5v + -12 + -5v Combine like terms: 7 + -12 = -5 -5 + -54v + -5v + -18v2 = 12 + 5v + -12 + -5v Combine like terms: -54v + -5v = -59v -5 + -59v + -18v2 = 12 + 5v + -12 + -5v Reorder the terms: -5 + -59v + -18v2 = 12 + -12 + 5v + -5v Combine like terms: 12 + -12 = 0 -5 + -59v + -18v2 = 0 + 5v + -5v -5 + -59v + -18v2 = 5v + -5v Combine like terms: 5v + -5v = 0 -5 + -59v + -18v2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(5 + 59v + 18v2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(5 + 59v + 18v2)' equal to zero and attempt to solve: Simplifying 5 + 59v + 18v2 = 0 Solving 5 + 59v + 18v2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.2777777778 + 3.277777778v + v2 = 0 Move the constant term to the right: Add '-0.2777777778' to each side of the equation. 0.2777777778 + 3.277777778v + -0.2777777778 + v2 = 0 + -0.2777777778 Reorder the terms: 0.2777777778 + -0.2777777778 + 3.277777778v + v2 = 0 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + 3.277777778v + v2 = 0 + -0.2777777778 3.277777778v + v2 = 0 + -0.2777777778 Combine like terms: 0 + -0.2777777778 = -0.2777777778 3.277777778v + v2 = -0.2777777778 The v term is 3.277777778v. Take half its coefficient (1.638888889). Square it (2.685956790) and add it to both sides. Add '2.685956790' to each side of the equation. 3.277777778v + 2.685956790 + v2 = -0.2777777778 + 2.685956790 Reorder the terms: 2.685956790 + 3.277777778v + v2 = -0.2777777778 + 2.685956790 Combine like terms: -0.2777777778 + 2.685956790 = 2.4081790122 2.685956790 + 3.277777778v + v2 = 2.4081790122 Factor a perfect square on the left side: (v + 1.638888889)(v + 1.638888889) = 2.4081790122 Calculate the square root of the right side: 1.551830858 Break this problem into two subproblems by setting (v + 1.638888889) equal to 1.551830858 and -1.551830858.Subproblem 1
v + 1.638888889 = 1.551830858 Simplifying v + 1.638888889 = 1.551830858 Reorder the terms: 1.638888889 + v = 1.551830858 Solving 1.638888889 + v = 1.551830858 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.638888889' to each side of the equation. 1.638888889 + -1.638888889 + v = 1.551830858 + -1.638888889 Combine like terms: 1.638888889 + -1.638888889 = 0.000000000 0.000000000 + v = 1.551830858 + -1.638888889 v = 1.551830858 + -1.638888889 Combine like terms: 1.551830858 + -1.638888889 = -0.087058031 v = -0.087058031 Simplifying v = -0.087058031Subproblem 2
v + 1.638888889 = -1.551830858 Simplifying v + 1.638888889 = -1.551830858 Reorder the terms: 1.638888889 + v = -1.551830858 Solving 1.638888889 + v = -1.551830858 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.638888889' to each side of the equation. 1.638888889 + -1.638888889 + v = -1.551830858 + -1.638888889 Combine like terms: 1.638888889 + -1.638888889 = 0.000000000 0.000000000 + v = -1.551830858 + -1.638888889 v = -1.551830858 + -1.638888889 Combine like terms: -1.551830858 + -1.638888889 = -3.190719747 v = -3.190719747 Simplifying v = -3.190719747Solution
The solution to the problem is based on the solutions from the subproblems. v = {-0.087058031, -3.190719747}Solution
v = {-0.087058031, -3.190719747}
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