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Simplifying 3v2 + 9v + 5 = 0 Reorder the terms: 5 + 9v + 3v2 = 0 Solving 5 + 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'. 1.666666667 + 3v + v2 = 0 Move the constant term to the right: Add '-1.666666667' to each side of the equation. 1.666666667 + 3v + -1.666666667 + v2 = 0 + -1.666666667 Reorder the terms: 1.666666667 + -1.666666667 + 3v + v2 = 0 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + 3v + v2 = 0 + -1.666666667 3v + v2 = 0 + -1.666666667 Combine like terms: 0 + -1.666666667 = -1.666666667 3v + v2 = -1.666666667 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 = -1.666666667 + 2.25 Reorder the terms: 2.25 + 3v + v2 = -1.666666667 + 2.25 Combine like terms: -1.666666667 + 2.25 = 0.583333333 2.25 + 3v + v2 = 0.583333333 Factor a perfect square on the left side: (v + 1.5)(v + 1.5) = 0.583333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (v + 1.5) equal to 0.763762616 and -0.763762616.Subproblem 1
v + 1.5 = 0.763762616 Simplifying v + 1.5 = 0.763762616 Reorder the terms: 1.5 + v = 0.763762616 Solving 1.5 + v = 0.763762616 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 = 0.763762616 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + v = 0.763762616 + -1.5 v = 0.763762616 + -1.5 Combine like terms: 0.763762616 + -1.5 = -0.736237384 v = -0.736237384 Simplifying v = -0.736237384Subproblem 2
v + 1.5 = -0.763762616 Simplifying v + 1.5 = -0.763762616 Reorder the terms: 1.5 + v = -0.763762616 Solving 1.5 + v = -0.763762616 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 = -0.763762616 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + v = -0.763762616 + -1.5 v = -0.763762616 + -1.5 Combine like terms: -0.763762616 + -1.5 = -2.263762616 v = -2.263762616 Simplifying v = -2.263762616Solution
The solution to the problem is based on the solutions from the subproblems. v = {-0.736237384, -2.263762616}
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