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Simplifying 5v2 + 9v + -8 + v + -3 + 14v2 + -4v + 6 = 0 Reorder the terms: -8 + -3 + 6 + 9v + v + -4v + 5v2 + 14v2 = 0 Combine like terms: -8 + -3 = -11 -11 + 6 + 9v + v + -4v + 5v2 + 14v2 = 0 Combine like terms: -11 + 6 = -5 -5 + 9v + v + -4v + 5v2 + 14v2 = 0 Combine like terms: 9v + v = 10v -5 + 10v + -4v + 5v2 + 14v2 = 0 Combine like terms: 10v + -4v = 6v -5 + 6v + 5v2 + 14v2 = 0 Combine like terms: 5v2 + 14v2 = 19v2 -5 + 6v + 19v2 = 0 Solving -5 + 6v + 19v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 19 the coefficient of the squared term: Divide each side by '19'. -0.2631578947 + 0.3157894737v + v2 = 0 Move the constant term to the right: Add '0.2631578947' to each side of the equation. -0.2631578947 + 0.3157894737v + 0.2631578947 + v2 = 0 + 0.2631578947 Reorder the terms: -0.2631578947 + 0.2631578947 + 0.3157894737v + v2 = 0 + 0.2631578947 Combine like terms: -0.2631578947 + 0.2631578947 = 0.0000000000 0.0000000000 + 0.3157894737v + v2 = 0 + 0.2631578947 0.3157894737v + v2 = 0 + 0.2631578947 Combine like terms: 0 + 0.2631578947 = 0.2631578947 0.3157894737v + v2 = 0.2631578947 The v term is 0.3157894737v. Take half its coefficient (0.1578947369). Square it (0.02493074794) and add it to both sides. Add '0.02493074794' to each side of the equation. 0.3157894737v + 0.02493074794 + v2 = 0.2631578947 + 0.02493074794 Reorder the terms: 0.02493074794 + 0.3157894737v + v2 = 0.2631578947 + 0.02493074794 Combine like terms: 0.2631578947 + 0.02493074794 = 0.28808864264 0.02493074794 + 0.3157894737v + v2 = 0.28808864264 Factor a perfect square on the left side: (v + 0.1578947369)(v + 0.1578947369) = 0.28808864264 Calculate the square root of the right side: 0.536738896 Break this problem into two subproblems by setting (v + 0.1578947369) equal to 0.536738896 and -0.536738896.Subproblem 1
v + 0.1578947369 = 0.536738896 Simplifying v + 0.1578947369 = 0.536738896 Reorder the terms: 0.1578947369 + v = 0.536738896 Solving 0.1578947369 + v = 0.536738896 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.1578947369' to each side of the equation. 0.1578947369 + -0.1578947369 + v = 0.536738896 + -0.1578947369 Combine like terms: 0.1578947369 + -0.1578947369 = 0.0000000000 0.0000000000 + v = 0.536738896 + -0.1578947369 v = 0.536738896 + -0.1578947369 Combine like terms: 0.536738896 + -0.1578947369 = 0.3788441591 v = 0.3788441591 Simplifying v = 0.3788441591Subproblem 2
v + 0.1578947369 = -0.536738896 Simplifying v + 0.1578947369 = -0.536738896 Reorder the terms: 0.1578947369 + v = -0.536738896 Solving 0.1578947369 + v = -0.536738896 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.1578947369' to each side of the equation. 0.1578947369 + -0.1578947369 + v = -0.536738896 + -0.1578947369 Combine like terms: 0.1578947369 + -0.1578947369 = 0.0000000000 0.0000000000 + v = -0.536738896 + -0.1578947369 v = -0.536738896 + -0.1578947369 Combine like terms: -0.536738896 + -0.1578947369 = -0.6946336329 v = -0.6946336329 Simplifying v = -0.6946336329Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.3788441591, -0.6946336329}
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