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Simplifying 3v2 + 9v + 4 = 0 Reorder the terms: 4 + 9v + 3v2 = 0 Solving 4 + 9v + 3v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + 3v + v2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + 3v + -1.333333333 + v2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + 3v + v2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + 3v + v2 = 0 + -1.333333333 3v + v2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 3v + v2 = -1.333333333 The v term is 3v. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3v + 2.25 + v2 = -1.333333333 + 2.25 Reorder the terms: 2.25 + 3v + v2 = -1.333333333 + 2.25 Combine like terms: -1.333333333 + 2.25 = 0.916666667 2.25 + 3v + v2 = 0.916666667 Factor a perfect square on the left side: (v + 1.5)(v + 1.5) = 0.916666667 Calculate the square root of the right side: 0.957427108 Break this problem into two subproblems by setting (v + 1.5) equal to 0.957427108 and -0.957427108.Subproblem 1
v + 1.5 = 0.957427108 Simplifying v + 1.5 = 0.957427108 Reorder the terms: 1.5 + v = 0.957427108 Solving 1.5 + v = 0.957427108 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + v = 0.957427108 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + v = 0.957427108 + -1.5 v = 0.957427108 + -1.5 Combine like terms: 0.957427108 + -1.5 = -0.542572892 v = -0.542572892 Simplifying v = -0.542572892Subproblem 2
v + 1.5 = -0.957427108 Simplifying v + 1.5 = -0.957427108 Reorder the terms: 1.5 + v = -0.957427108 Solving 1.5 + v = -0.957427108 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + v = -0.957427108 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + v = -0.957427108 + -1.5 v = -0.957427108 + -1.5 Combine like terms: -0.957427108 + -1.5 = -2.457427108 v = -2.457427108 Simplifying v = -2.457427108Solution
The solution to the problem is based on the solutions from the subproblems. v = {-0.542572892, -2.457427108}
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