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Simplifying (v + -7) + (5v2 + 12v + -7) + (13v2 + -7v + 9) = 0 Reorder the terms: (-7 + v) + (5v2 + 12v + -7) + (13v2 + -7v + 9) = 0 Remove parenthesis around (-7 + v) -7 + v + (5v2 + 12v + -7) + (13v2 + -7v + 9) = 0 Reorder the terms: -7 + v + (-7 + 12v + 5v2) + (13v2 + -7v + 9) = 0 Remove parenthesis around (-7 + 12v + 5v2) -7 + v + -7 + 12v + 5v2 + (13v2 + -7v + 9) = 0 Reorder the terms: -7 + v + -7 + 12v + 5v2 + (9 + -7v + 13v2) = 0 Remove parenthesis around (9 + -7v + 13v2) -7 + v + -7 + 12v + 5v2 + 9 + -7v + 13v2 = 0 Reorder the terms: -7 + -7 + 9 + v + 12v + -7v + 5v2 + 13v2 = 0 Combine like terms: -7 + -7 = -14 -14 + 9 + v + 12v + -7v + 5v2 + 13v2 = 0 Combine like terms: -14 + 9 = -5 -5 + v + 12v + -7v + 5v2 + 13v2 = 0 Combine like terms: v + 12v = 13v -5 + 13v + -7v + 5v2 + 13v2 = 0 Combine like terms: 13v + -7v = 6v -5 + 6v + 5v2 + 13v2 = 0 Combine like terms: 5v2 + 13v2 = 18v2 -5 + 6v + 18v2 = 0 Solving -5 + 6v + 18v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -0.2777777778 + 0.3333333333v + v2 = 0 Move the constant term to the right: Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.3333333333v + 0.2777777778 + v2 = 0 + 0.2777777778 Reorder the terms: -0.2777777778 + 0.2777777778 + 0.3333333333v + v2 = 0 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + 0.3333333333v + v2 = 0 + 0.2777777778 0.3333333333v + v2 = 0 + 0.2777777778 Combine like terms: 0 + 0.2777777778 = 0.2777777778 0.3333333333v + v2 = 0.2777777778 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 = 0.2777777778 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333v + v2 = 0.2777777778 + 0.02777777779 Combine like terms: 0.2777777778 + 0.02777777779 = 0.30555555559 0.02777777779 + 0.3333333333v + v2 = 0.30555555559 Factor a perfect square on the left side: (v + 0.1666666667)(v + 0.1666666667) = 0.30555555559 Calculate the square root of the right side: 0.552770798 Break this problem into two subproblems by setting (v + 0.1666666667) equal to 0.552770798 and -0.552770798.Subproblem 1
v + 0.1666666667 = 0.552770798 Simplifying v + 0.1666666667 = 0.552770798 Reorder the terms: 0.1666666667 + v = 0.552770798 Solving 0.1666666667 + v = 0.552770798 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 = 0.552770798 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + v = 0.552770798 + -0.1666666667 v = 0.552770798 + -0.1666666667 Combine like terms: 0.552770798 + -0.1666666667 = 0.3861041313 v = 0.3861041313 Simplifying v = 0.3861041313Subproblem 2
v + 0.1666666667 = -0.552770798 Simplifying v + 0.1666666667 = -0.552770798 Reorder the terms: 0.1666666667 + v = -0.552770798 Solving 0.1666666667 + v = -0.552770798 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 = -0.552770798 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + v = -0.552770798 + -0.1666666667 v = -0.552770798 + -0.1666666667 Combine like terms: -0.552770798 + -0.1666666667 = -0.7194374647 v = -0.7194374647 Simplifying v = -0.7194374647Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.3861041313, -0.7194374647}
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