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Simplifying 5(v + 2) * 3(v + 5) = 1 Reorder the terms: 5(2 + v) * 3(v + 5) = 1 Reorder the terms: 5(2 + v) * 3(5 + v) = 1 Reorder the terms for easier multiplication: 5 * 3(2 + v)(5 + v) = 1 Multiply 5 * 3 15(2 + v)(5 + v) = 1 Multiply (2 + v) * (5 + v) 15(2(5 + v) + v(5 + v)) = 1 15((5 * 2 + v * 2) + v(5 + v)) = 1 15((10 + 2v) + v(5 + v)) = 1 15(10 + 2v + (5 * v + v * v)) = 1 15(10 + 2v + (5v + v2)) = 1 Combine like terms: 2v + 5v = 7v 15(10 + 7v + v2) = 1 (10 * 15 + 7v * 15 + v2 * 15) = 1 (150 + 105v + 15v2) = 1 Solving 150 + 105v + 15v2 = 1 Solving for variable 'v'. Reorder the terms: 150 + -1 + 105v + 15v2 = 1 + -1 Combine like terms: 150 + -1 = 149 149 + 105v + 15v2 = 1 + -1 Combine like terms: 1 + -1 = 0 149 + 105v + 15v2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. 9.933333333 + 7v + v2 = 0 Move the constant term to the right: Add '-9.933333333' to each side of the equation. 9.933333333 + 7v + -9.933333333 + v2 = 0 + -9.933333333 Reorder the terms: 9.933333333 + -9.933333333 + 7v + v2 = 0 + -9.933333333 Combine like terms: 9.933333333 + -9.933333333 = 0.000000000 0.000000000 + 7v + v2 = 0 + -9.933333333 7v + v2 = 0 + -9.933333333 Combine like terms: 0 + -9.933333333 = -9.933333333 7v + v2 = -9.933333333 The v term is 7v. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7v + 12.25 + v2 = -9.933333333 + 12.25 Reorder the terms: 12.25 + 7v + v2 = -9.933333333 + 12.25 Combine like terms: -9.933333333 + 12.25 = 2.316666667 12.25 + 7v + v2 = 2.316666667 Factor a perfect square on the left side: (v + 3.5)(v + 3.5) = 2.316666667 Calculate the square root of the right side: 1.522060008 Break this problem into two subproblems by setting (v + 3.5) equal to 1.522060008 and -1.522060008.Subproblem 1
v + 3.5 = 1.522060008 Simplifying v + 3.5 = 1.522060008 Reorder the terms: 3.5 + v = 1.522060008 Solving 3.5 + v = 1.522060008 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + v = 1.522060008 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + v = 1.522060008 + -3.5 v = 1.522060008 + -3.5 Combine like terms: 1.522060008 + -3.5 = -1.977939992 v = -1.977939992 Simplifying v = -1.977939992Subproblem 2
v + 3.5 = -1.522060008 Simplifying v + 3.5 = -1.522060008 Reorder the terms: 3.5 + v = -1.522060008 Solving 3.5 + v = -1.522060008 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + v = -1.522060008 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + v = -1.522060008 + -3.5 v = -1.522060008 + -3.5 Combine like terms: -1.522060008 + -3.5 = -5.022060008 v = -5.022060008 Simplifying v = -5.022060008Solution
The solution to the problem is based on the solutions from the subproblems. v = {-1.977939992, -5.022060008}
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