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Simplifying 13v2 + 27v + 8 = 0 Reorder the terms: 8 + 27v + 13v2 = 0 Solving 8 + 27v + 13v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. 0.6153846154 + 2.076923077v + v2 = 0 Move the constant term to the right: Add '-0.6153846154' to each side of the equation. 0.6153846154 + 2.076923077v + -0.6153846154 + v2 = 0 + -0.6153846154 Reorder the terms: 0.6153846154 + -0.6153846154 + 2.076923077v + v2 = 0 + -0.6153846154 Combine like terms: 0.6153846154 + -0.6153846154 = 0.0000000000 0.0000000000 + 2.076923077v + v2 = 0 + -0.6153846154 2.076923077v + v2 = 0 + -0.6153846154 Combine like terms: 0 + -0.6153846154 = -0.6153846154 2.076923077v + v2 = -0.6153846154 The v term is 2.076923077v. Take half its coefficient (1.038461539). Square it (1.078402368) and add it to both sides. Add '1.078402368' to each side of the equation. 2.076923077v + 1.078402368 + v2 = -0.6153846154 + 1.078402368 Reorder the terms: 1.078402368 + 2.076923077v + v2 = -0.6153846154 + 1.078402368 Combine like terms: -0.6153846154 + 1.078402368 = 0.4630177526 1.078402368 + 2.076923077v + v2 = 0.4630177526 Factor a perfect square on the left side: (v + 1.038461539)(v + 1.038461539) = 0.4630177526 Calculate the square root of the right side: 0.680454078 Break this problem into two subproblems by setting (v + 1.038461539) equal to 0.680454078 and -0.680454078.Subproblem 1
v + 1.038461539 = 0.680454078 Simplifying v + 1.038461539 = 0.680454078 Reorder the terms: 1.038461539 + v = 0.680454078 Solving 1.038461539 + v = 0.680454078 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.038461539' to each side of the equation. 1.038461539 + -1.038461539 + v = 0.680454078 + -1.038461539 Combine like terms: 1.038461539 + -1.038461539 = 0.000000000 0.000000000 + v = 0.680454078 + -1.038461539 v = 0.680454078 + -1.038461539 Combine like terms: 0.680454078 + -1.038461539 = -0.358007461 v = -0.358007461 Simplifying v = -0.358007461Subproblem 2
v + 1.038461539 = -0.680454078 Simplifying v + 1.038461539 = -0.680454078 Reorder the terms: 1.038461539 + v = -0.680454078 Solving 1.038461539 + v = -0.680454078 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.038461539' to each side of the equation. 1.038461539 + -1.038461539 + v = -0.680454078 + -1.038461539 Combine like terms: 1.038461539 + -1.038461539 = 0.000000000 0.000000000 + v = -0.680454078 + -1.038461539 v = -0.680454078 + -1.038461539 Combine like terms: -0.680454078 + -1.038461539 = -1.718915617 v = -1.718915617 Simplifying v = -1.718915617Solution
The solution to the problem is based on the solutions from the subproblems. v = {-0.358007461, -1.718915617}
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