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Simplifying -2v2 + -1v + 12 = 13v2 + 6v Reorder the terms: 12 + -1v + -2v2 = 13v2 + 6v Reorder the terms: 12 + -1v + -2v2 = 6v + 13v2 Solving 12 + -1v + -2v2 = 6v + 13v2 Solving for variable 'v'. Reorder the terms: 12 + -1v + -6v + -2v2 + -13v2 = 6v + 13v2 + -6v + -13v2 Combine like terms: -1v + -6v = -7v 12 + -7v + -2v2 + -13v2 = 6v + 13v2 + -6v + -13v2 Combine like terms: -2v2 + -13v2 = -15v2 12 + -7v + -15v2 = 6v + 13v2 + -6v + -13v2 Reorder the terms: 12 + -7v + -15v2 = 6v + -6v + 13v2 + -13v2 Combine like terms: 6v + -6v = 0 12 + -7v + -15v2 = 0 + 13v2 + -13v2 12 + -7v + -15v2 = 13v2 + -13v2 Combine like terms: 13v2 + -13v2 = 0 12 + -7v + -15v2 = 0 Begin completing the square. Divide all terms by -15 the coefficient of the squared term: Divide each side by '-15'. -0.8 + 0.4666666667v + v2 = 0 Move the constant term to the right: Add '0.8' to each side of the equation. -0.8 + 0.4666666667v + 0.8 + v2 = 0 + 0.8 Reorder the terms: -0.8 + 0.8 + 0.4666666667v + v2 = 0 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + 0.4666666667v + v2 = 0 + 0.8 0.4666666667v + v2 = 0 + 0.8 Combine like terms: 0 + 0.8 = 0.8 0.4666666667v + v2 = 0.8 The v term is 0.4666666667v. Take half its coefficient (0.2333333334). Square it (0.05444444448) and add it to both sides. Add '0.05444444448' to each side of the equation. 0.4666666667v + 0.05444444448 + v2 = 0.8 + 0.05444444448 Reorder the terms: 0.05444444448 + 0.4666666667v + v2 = 0.8 + 0.05444444448 Combine like terms: 0.8 + 0.05444444448 = 0.85444444448 0.05444444448 + 0.4666666667v + v2 = 0.85444444448 Factor a perfect square on the left side: (v + 0.2333333334)(v + 0.2333333334) = 0.85444444448 Calculate the square root of the right side: 0.924361642 Break this problem into two subproblems by setting (v + 0.2333333334) equal to 0.924361642 and -0.924361642.Subproblem 1
v + 0.2333333334 = 0.924361642 Simplifying v + 0.2333333334 = 0.924361642 Reorder the terms: 0.2333333334 + v = 0.924361642 Solving 0.2333333334 + v = 0.924361642 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.2333333334' to each side of the equation. 0.2333333334 + -0.2333333334 + v = 0.924361642 + -0.2333333334 Combine like terms: 0.2333333334 + -0.2333333334 = 0.0000000000 0.0000000000 + v = 0.924361642 + -0.2333333334 v = 0.924361642 + -0.2333333334 Combine like terms: 0.924361642 + -0.2333333334 = 0.6910283086 v = 0.6910283086 Simplifying v = 0.6910283086Subproblem 2
v + 0.2333333334 = -0.924361642 Simplifying v + 0.2333333334 = -0.924361642 Reorder the terms: 0.2333333334 + v = -0.924361642 Solving 0.2333333334 + v = -0.924361642 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.2333333334' to each side of the equation. 0.2333333334 + -0.2333333334 + v = -0.924361642 + -0.2333333334 Combine like terms: 0.2333333334 + -0.2333333334 = 0.0000000000 0.0000000000 + v = -0.924361642 + -0.2333333334 v = -0.924361642 + -0.2333333334 Combine like terms: -0.924361642 + -0.2333333334 = -1.1576949754 v = -1.1576949754 Simplifying v = -1.1576949754Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.6910283086, -1.1576949754}
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