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Simplifying 0 = 6v2 + 5v + -9 Reorder the terms: 0 = -9 + 5v + 6v2 Solving 0 = -9 + 5v + 6v2 Solving for variable 'v'. Combine like terms: 0 + 9 = 9 9 + -5v + -6v2 = -9 + 5v + 6v2 + 9 + -5v + -6v2 Reorder the terms: 9 + -5v + -6v2 = -9 + 9 + 5v + -5v + 6v2 + -6v2 Combine like terms: -9 + 9 = 0 9 + -5v + -6v2 = 0 + 5v + -5v + 6v2 + -6v2 9 + -5v + -6v2 = 5v + -5v + 6v2 + -6v2 Combine like terms: 5v + -5v = 0 9 + -5v + -6v2 = 0 + 6v2 + -6v2 9 + -5v + -6v2 = 6v2 + -6v2 Combine like terms: 6v2 + -6v2 = 0 9 + -5v + -6v2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1.5 + 0.8333333333v + v2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 0.8333333333v + 1.5 + v2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 0.8333333333v + v2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 0.8333333333v + v2 = 0 + 1.5 0.8333333333v + v2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 0.8333333333v + v2 = 1.5 The v term is 0.8333333333v. Take half its coefficient (0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. 0.8333333333v + 0.1736111111 + v2 = 1.5 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333v + v2 = 1.5 + 0.1736111111 Combine like terms: 1.5 + 0.1736111111 = 1.6736111111 0.1736111111 + 0.8333333333v + v2 = 1.6736111111 Factor a perfect square on the left side: (v + 0.4166666667)(v + 0.4166666667) = 1.6736111111 Calculate the square root of the right side: 1.293681225 Break this problem into two subproblems by setting (v + 0.4166666667) equal to 1.293681225 and -1.293681225.Subproblem 1
v + 0.4166666667 = 1.293681225 Simplifying v + 0.4166666667 = 1.293681225 Reorder the terms: 0.4166666667 + v = 1.293681225 Solving 0.4166666667 + v = 1.293681225 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + v = 1.293681225 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + v = 1.293681225 + -0.4166666667 v = 1.293681225 + -0.4166666667 Combine like terms: 1.293681225 + -0.4166666667 = 0.8770145583 v = 0.8770145583 Simplifying v = 0.8770145583Subproblem 2
v + 0.4166666667 = -1.293681225 Simplifying v + 0.4166666667 = -1.293681225 Reorder the terms: 0.4166666667 + v = -1.293681225 Solving 0.4166666667 + v = -1.293681225 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + v = -1.293681225 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + v = -1.293681225 + -0.4166666667 v = -1.293681225 + -0.4166666667 Combine like terms: -1.293681225 + -0.4166666667 = -1.7103478917 v = -1.7103478917 Simplifying v = -1.7103478917Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.8770145583, -1.7103478917}
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