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Simplifying 0 = 8v2 + 4v + -7 + v2 + -1v Reorder the terms: 0 = -7 + 4v + -1v + 8v2 + v2 Combine like terms: 4v + -1v = 3v 0 = -7 + 3v + 8v2 + v2 Combine like terms: 8v2 + v2 = 9v2 0 = -7 + 3v + 9v2 Solving 0 = -7 + 3v + 9v2 Solving for variable 'v'. Combine like terms: 0 + 7 = 7 7 + -3v + -9v2 = -7 + 3v + 9v2 + 7 + -3v + -9v2 Reorder the terms: 7 + -3v + -9v2 = -7 + 7 + 3v + -3v + 9v2 + -9v2 Combine like terms: -7 + 7 = 0 7 + -3v + -9v2 = 0 + 3v + -3v + 9v2 + -9v2 7 + -3v + -9v2 = 3v + -3v + 9v2 + -9v2 Combine like terms: 3v + -3v = 0 7 + -3v + -9v2 = 0 + 9v2 + -9v2 7 + -3v + -9v2 = 9v2 + -9v2 Combine like terms: 9v2 + -9v2 = 0 7 + -3v + -9v2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.7777777778 + 0.3333333333v + v2 = 0 Move the constant term to the right: Add '0.7777777778' to each side of the equation. -0.7777777778 + 0.3333333333v + 0.7777777778 + v2 = 0 + 0.7777777778 Reorder the terms: -0.7777777778 + 0.7777777778 + 0.3333333333v + v2 = 0 + 0.7777777778 Combine like terms: -0.7777777778 + 0.7777777778 = 0.0000000000 0.0000000000 + 0.3333333333v + v2 = 0 + 0.7777777778 0.3333333333v + v2 = 0 + 0.7777777778 Combine like terms: 0 + 0.7777777778 = 0.7777777778 0.3333333333v + v2 = 0.7777777778 The v term is 0.3333333333v. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333v + 0.02777777779 + v2 = 0.7777777778 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333v + v2 = 0.7777777778 + 0.02777777779 Combine like terms: 0.7777777778 + 0.02777777779 = 0.80555555559 0.02777777779 + 0.3333333333v + v2 = 0.80555555559 Factor a perfect square on the left side: (v + 0.1666666667)(v + 0.1666666667) = 0.80555555559 Calculate the square root of the right side: 0.897527468 Break this problem into two subproblems by setting (v + 0.1666666667) equal to 0.897527468 and -0.897527468.Subproblem 1
v + 0.1666666667 = 0.897527468 Simplifying v + 0.1666666667 = 0.897527468 Reorder the terms: 0.1666666667 + v = 0.897527468 Solving 0.1666666667 + v = 0.897527468 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + v = 0.897527468 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + v = 0.897527468 + -0.1666666667 v = 0.897527468 + -0.1666666667 Combine like terms: 0.897527468 + -0.1666666667 = 0.7308608013 v = 0.7308608013 Simplifying v = 0.7308608013Subproblem 2
v + 0.1666666667 = -0.897527468 Simplifying v + 0.1666666667 = -0.897527468 Reorder the terms: 0.1666666667 + v = -0.897527468 Solving 0.1666666667 + v = -0.897527468 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + v = -0.897527468 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + v = -0.897527468 + -0.1666666667 v = -0.897527468 + -0.1666666667 Combine like terms: -0.897527468 + -0.1666666667 = -1.0641941347 v = -1.0641941347 Simplifying v = -1.0641941347Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.7308608013, -1.0641941347}
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