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Simplifying 6(v + -6) * 8v = -32 Reorder the terms: 6(-6 + v) * 8v = -32 Reorder the terms for easier multiplication: 6 * 8v(-6 + v) = -32 Multiply 6 * 8 48v(-6 + v) = -32 (-6 * 48v + v * 48v) = -32 (-288v + 48v2) = -32 Solving -288v + 48v2 = -32 Solving for variable 'v'. Reorder the terms: 32 + -288v + 48v2 = -32 + 32 Combine like terms: -32 + 32 = 0 32 + -288v + 48v2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(2 + -18v + 3v2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(2 + -18v + 3v2)' equal to zero and attempt to solve: Simplifying 2 + -18v + 3v2 = 0 Solving 2 + -18v + 3v2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.6666666667 + -6v + v2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + -6v + -0.6666666667 + v2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + -6v + v2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + -6v + v2 = 0 + -0.6666666667 -6v + v2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 -6v + v2 = -0.6666666667 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 = -0.6666666667 + 9 Reorder the terms: 9 + -6v + v2 = -0.6666666667 + 9 Combine like terms: -0.6666666667 + 9 = 8.3333333333 9 + -6v + v2 = 8.3333333333 Factor a perfect square on the left side: (v + -3)(v + -3) = 8.3333333333 Calculate the square root of the right side: 2.886751346 Break this problem into two subproblems by setting (v + -3) equal to 2.886751346 and -2.886751346.Subproblem 1
v + -3 = 2.886751346 Simplifying v + -3 = 2.886751346 Reorder the terms: -3 + v = 2.886751346 Solving -3 + v = 2.886751346 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 = 2.886751346 + 3 Combine like terms: -3 + 3 = 0 0 + v = 2.886751346 + 3 v = 2.886751346 + 3 Combine like terms: 2.886751346 + 3 = 5.886751346 v = 5.886751346 Simplifying v = 5.886751346Subproblem 2
v + -3 = -2.886751346 Simplifying v + -3 = -2.886751346 Reorder the terms: -3 + v = -2.886751346 Solving -3 + v = -2.886751346 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 = -2.886751346 + 3 Combine like terms: -3 + 3 = 0 0 + v = -2.886751346 + 3 v = -2.886751346 + 3 Combine like terms: -2.886751346 + 3 = 0.113248654 v = 0.113248654 Simplifying v = 0.113248654Solution
The solution to the problem is based on the solutions from the subproblems. v = {5.886751346, 0.113248654}Solution
v = {5.886751346, 0.113248654}
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