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Simplifying 3v2 + 18v + 27 = 81 Reorder the terms: 27 + 18v + 3v2 = 81 Solving 27 + 18v + 3v2 = 81 Solving for variable 'v'. Reorder the terms: 27 + -81 + 18v + 3v2 = 81 + -81 Combine like terms: 27 + -81 = -54 -54 + 18v + 3v2 = 81 + -81 Combine like terms: 81 + -81 = 0 -54 + 18v + 3v2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-18 + 6v + v2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-18 + 6v + v2)' equal to zero and attempt to solve: Simplifying -18 + 6v + v2 = 0 Solving -18 + 6v + v2 = 0 Begin completing the square. Move the constant term to the right: Add '18' to each side of the equation. -18 + 6v + 18 + v2 = 0 + 18 Reorder the terms: -18 + 18 + 6v + v2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 6v + v2 = 0 + 18 6v + v2 = 0 + 18 Combine like terms: 0 + 18 = 18 6v + v2 = 18 The v term is 6v. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6v + 9 + v2 = 18 + 9 Reorder the terms: 9 + 6v + v2 = 18 + 9 Combine like terms: 18 + 9 = 27 9 + 6v + v2 = 27 Factor a perfect square on the left side: (v + 3)(v + 3) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (v + 3) equal to 5.196152423 and -5.196152423.Subproblem 1
v + 3 = 5.196152423 Simplifying v + 3 = 5.196152423 Reorder the terms: 3 + v = 5.196152423 Solving 3 + v = 5.196152423 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + v = 5.196152423 + -3 Combine like terms: 3 + -3 = 0 0 + v = 5.196152423 + -3 v = 5.196152423 + -3 Combine like terms: 5.196152423 + -3 = 2.196152423 v = 2.196152423 Simplifying v = 2.196152423Subproblem 2
v + 3 = -5.196152423 Simplifying v + 3 = -5.196152423 Reorder the terms: 3 + v = -5.196152423 Solving 3 + v = -5.196152423 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + v = -5.196152423 + -3 Combine like terms: 3 + -3 = 0 0 + v = -5.196152423 + -3 v = -5.196152423 + -3 Combine like terms: -5.196152423 + -3 = -8.196152423 v = -8.196152423 Simplifying v = -8.196152423Solution
The solution to the problem is based on the solutions from the subproblems. v = {2.196152423, -8.196152423}Solution
v = {2.196152423, -8.196152423}
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