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Simplifying 3v2 + 12v = -8 Reorder the terms: 12v + 3v2 = -8 Solving 12v + 3v2 = -8 Solving for variable 'v'. Reorder the terms: 8 + 12v + 3v2 = -8 + 8 Combine like terms: -8 + 8 = 0 8 + 12v + 3v2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 2.666666667 + 4v + v2 = 0 Move the constant term to the right: Add '-2.666666667' to each side of the equation. 2.666666667 + 4v + -2.666666667 + v2 = 0 + -2.666666667 Reorder the terms: 2.666666667 + -2.666666667 + 4v + v2 = 0 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + 4v + v2 = 0 + -2.666666667 4v + v2 = 0 + -2.666666667 Combine like terms: 0 + -2.666666667 = -2.666666667 4v + v2 = -2.666666667 The v term is 4v. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4v + 4 + v2 = -2.666666667 + 4 Reorder the terms: 4 + 4v + v2 = -2.666666667 + 4 Combine like terms: -2.666666667 + 4 = 1.333333333 4 + 4v + v2 = 1.333333333 Factor a perfect square on the left side: (v + 2)(v + 2) = 1.333333333 Calculate the square root of the right side: 1.154700538 Break this problem into two subproblems by setting (v + 2) equal to 1.154700538 and -1.154700538.Subproblem 1
v + 2 = 1.154700538 Simplifying v + 2 = 1.154700538 Reorder the terms: 2 + v = 1.154700538 Solving 2 + v = 1.154700538 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + v = 1.154700538 + -2 Combine like terms: 2 + -2 = 0 0 + v = 1.154700538 + -2 v = 1.154700538 + -2 Combine like terms: 1.154700538 + -2 = -0.845299462 v = -0.845299462 Simplifying v = -0.845299462Subproblem 2
v + 2 = -1.154700538 Simplifying v + 2 = -1.154700538 Reorder the terms: 2 + v = -1.154700538 Solving 2 + v = -1.154700538 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + v = -1.154700538 + -2 Combine like terms: 2 + -2 = 0 0 + v = -1.154700538 + -2 v = -1.154700538 + -2 Combine like terms: -1.154700538 + -2 = -3.154700538 v = -3.154700538 Simplifying v = -3.154700538Solution
The solution to the problem is based on the solutions from the subproblems. v = {-0.845299462, -3.154700538}
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