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Simplifying 35v2 + 81v + -40 = 0 Reorder the terms: -40 + 81v + 35v2 = 0 Solving -40 + 81v + 35v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 35 the coefficient of the squared term: Divide each side by '35'. -1.142857143 + 2.314285714v + v2 = 0 Move the constant term to the right: Add '1.142857143' to each side of the equation. -1.142857143 + 2.314285714v + 1.142857143 + v2 = 0 + 1.142857143 Reorder the terms: -1.142857143 + 1.142857143 + 2.314285714v + v2 = 0 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + 2.314285714v + v2 = 0 + 1.142857143 2.314285714v + v2 = 0 + 1.142857143 Combine like terms: 0 + 1.142857143 = 1.142857143 2.314285714v + v2 = 1.142857143 The v term is 2.314285714v. Take half its coefficient (1.157142857). Square it (1.338979592) and add it to both sides. Add '1.338979592' to each side of the equation. 2.314285714v + 1.338979592 + v2 = 1.142857143 + 1.338979592 Reorder the terms: 1.338979592 + 2.314285714v + v2 = 1.142857143 + 1.338979592 Combine like terms: 1.142857143 + 1.338979592 = 2.481836735 1.338979592 + 2.314285714v + v2 = 2.481836735 Factor a perfect square on the left side: (v + 1.157142857)(v + 1.157142857) = 2.481836735 Calculate the square root of the right side: 1.575384631 Break this problem into two subproblems by setting (v + 1.157142857) equal to 1.575384631 and -1.575384631.Subproblem 1
v + 1.157142857 = 1.575384631 Simplifying v + 1.157142857 = 1.575384631 Reorder the terms: 1.157142857 + v = 1.575384631 Solving 1.157142857 + v = 1.575384631 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.157142857' to each side of the equation. 1.157142857 + -1.157142857 + v = 1.575384631 + -1.157142857 Combine like terms: 1.157142857 + -1.157142857 = 0.000000000 0.000000000 + v = 1.575384631 + -1.157142857 v = 1.575384631 + -1.157142857 Combine like terms: 1.575384631 + -1.157142857 = 0.418241774 v = 0.418241774 Simplifying v = 0.418241774Subproblem 2
v + 1.157142857 = -1.575384631 Simplifying v + 1.157142857 = -1.575384631 Reorder the terms: 1.157142857 + v = -1.575384631 Solving 1.157142857 + v = -1.575384631 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.157142857' to each side of the equation. 1.157142857 + -1.157142857 + v = -1.575384631 + -1.157142857 Combine like terms: 1.157142857 + -1.157142857 = 0.000000000 0.000000000 + v = -1.575384631 + -1.157142857 v = -1.575384631 + -1.157142857 Combine like terms: -1.575384631 + -1.157142857 = -2.732527488 v = -2.732527488 Simplifying v = -2.732527488Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.418241774, -2.732527488}
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