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Simplifying v(7v + -27) = -10 Reorder the terms: v(-27 + 7v) = -10 (-27 * v + 7v * v) = -10 (-27v + 7v2) = -10 Solving -27v + 7v2 = -10 Solving for variable 'v'. Reorder the terms: 10 + -27v + 7v2 = -10 + 10 Combine like terms: -10 + 10 = 0 10 + -27v + 7v2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 1.428571429 + -3.857142857v + v2 = 0 Move the constant term to the right: Add '-1.428571429' to each side of the equation. 1.428571429 + -3.857142857v + -1.428571429 + v2 = 0 + -1.428571429 Reorder the terms: 1.428571429 + -1.428571429 + -3.857142857v + v2 = 0 + -1.428571429 Combine like terms: 1.428571429 + -1.428571429 = 0.000000000 0.000000000 + -3.857142857v + v2 = 0 + -1.428571429 -3.857142857v + v2 = 0 + -1.428571429 Combine like terms: 0 + -1.428571429 = -1.428571429 -3.857142857v + v2 = -1.428571429 The v term is -3.857142857v. Take half its coefficient (-1.928571429). Square it (3.719387757) and add it to both sides. Add '3.719387757' to each side of the equation. -3.857142857v + 3.719387757 + v2 = -1.428571429 + 3.719387757 Reorder the terms: 3.719387757 + -3.857142857v + v2 = -1.428571429 + 3.719387757 Combine like terms: -1.428571429 + 3.719387757 = 2.290816328 3.719387757 + -3.857142857v + v2 = 2.290816328 Factor a perfect square on the left side: (v + -1.928571429)(v + -1.928571429) = 2.290816328 Calculate the square root of the right side: 1.513544293 Break this problem into two subproblems by setting (v + -1.928571429) equal to 1.513544293 and -1.513544293.Subproblem 1
v + -1.928571429 = 1.513544293 Simplifying v + -1.928571429 = 1.513544293 Reorder the terms: -1.928571429 + v = 1.513544293 Solving -1.928571429 + v = 1.513544293 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1.928571429' to each side of the equation. -1.928571429 + 1.928571429 + v = 1.513544293 + 1.928571429 Combine like terms: -1.928571429 + 1.928571429 = 0.000000000 0.000000000 + v = 1.513544293 + 1.928571429 v = 1.513544293 + 1.928571429 Combine like terms: 1.513544293 + 1.928571429 = 3.442115722 v = 3.442115722 Simplifying v = 3.442115722Subproblem 2
v + -1.928571429 = -1.513544293 Simplifying v + -1.928571429 = -1.513544293 Reorder the terms: -1.928571429 + v = -1.513544293 Solving -1.928571429 + v = -1.513544293 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1.928571429' to each side of the equation. -1.928571429 + 1.928571429 + v = -1.513544293 + 1.928571429 Combine like terms: -1.928571429 + 1.928571429 = 0.000000000 0.000000000 + v = -1.513544293 + 1.928571429 v = -1.513544293 + 1.928571429 Combine like terms: -1.513544293 + 1.928571429 = 0.415027136 v = 0.415027136 Simplifying v = 0.415027136Solution
The solution to the problem is based on the solutions from the subproblems. v = {3.442115722, 0.415027136}
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