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Simplifying 45v2 = 320 + -240v Solving 45v2 = 320 + -240v Solving for variable 'v'. Reorder the terms: -320 + 240v + 45v2 = 320 + -240v + -320 + 240v Reorder the terms: -320 + 240v + 45v2 = 320 + -320 + -240v + 240v Combine like terms: 320 + -320 = 0 -320 + 240v + 45v2 = 0 + -240v + 240v -320 + 240v + 45v2 = -240v + 240v Combine like terms: -240v + 240v = 0 -320 + 240v + 45v2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-64 + 48v + 9v2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-64 + 48v + 9v2)' equal to zero and attempt to solve: Simplifying -64 + 48v + 9v2 = 0 Solving -64 + 48v + 9v2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -7.111111111 + 5.333333333v + v2 = 0 Move the constant term to the right: Add '7.111111111' to each side of the equation. -7.111111111 + 5.333333333v + 7.111111111 + v2 = 0 + 7.111111111 Reorder the terms: -7.111111111 + 7.111111111 + 5.333333333v + v2 = 0 + 7.111111111 Combine like terms: -7.111111111 + 7.111111111 = 0.000000000 0.000000000 + 5.333333333v + v2 = 0 + 7.111111111 5.333333333v + v2 = 0 + 7.111111111 Combine like terms: 0 + 7.111111111 = 7.111111111 5.333333333v + v2 = 7.111111111 The v term is 5.333333333v. Take half its coefficient (2.666666667). Square it (7.111111113) and add it to both sides. Add '7.111111113' to each side of the equation. 5.333333333v + 7.111111113 + v2 = 7.111111111 + 7.111111113 Reorder the terms: 7.111111113 + 5.333333333v + v2 = 7.111111111 + 7.111111113 Combine like terms: 7.111111111 + 7.111111113 = 14.222222224 7.111111113 + 5.333333333v + v2 = 14.222222224 Factor a perfect square on the left side: (v + 2.666666667)(v + 2.666666667) = 14.222222224 Calculate the square root of the right side: 3.771236167 Break this problem into two subproblems by setting (v + 2.666666667) equal to 3.771236167 and -3.771236167.Subproblem 1
v + 2.666666667 = 3.771236167 Simplifying v + 2.666666667 = 3.771236167 Reorder the terms: 2.666666667 + v = 3.771236167 Solving 2.666666667 + v = 3.771236167 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + v = 3.771236167 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + v = 3.771236167 + -2.666666667 v = 3.771236167 + -2.666666667 Combine like terms: 3.771236167 + -2.666666667 = 1.1045695 v = 1.1045695 Simplifying v = 1.1045695Subproblem 2
v + 2.666666667 = -3.771236167 Simplifying v + 2.666666667 = -3.771236167 Reorder the terms: 2.666666667 + v = -3.771236167 Solving 2.666666667 + v = -3.771236167 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + v = -3.771236167 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + v = -3.771236167 + -2.666666667 v = -3.771236167 + -2.666666667 Combine like terms: -3.771236167 + -2.666666667 = -6.437902834 v = -6.437902834 Simplifying v = -6.437902834Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.1045695, -6.437902834}Solution
v = {1.1045695, -6.437902834}
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