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Simplifying 9v2 + 9v + -23 = 0 Reorder the terms: -23 + 9v + 9v2 = 0 Solving -23 + 9v + 9v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.555555556 + v + v2 = 0 Move the constant term to the right: Add '2.555555556' to each side of the equation. -2.555555556 + v + 2.555555556 + v2 = 0 + 2.555555556 Reorder the terms: -2.555555556 + 2.555555556 + v + v2 = 0 + 2.555555556 Combine like terms: -2.555555556 + 2.555555556 = 0.000000000 0.000000000 + v + v2 = 0 + 2.555555556 v + v2 = 0 + 2.555555556 Combine like terms: 0 + 2.555555556 = 2.555555556 v + v2 = 2.555555556 The v term is v. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + v2 = 2.555555556 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + v2 = 2.555555556 + 0.25 Combine like terms: 2.555555556 + 0.25 = 2.805555556 1.25 + v2 = 2.805555556 Factor a perfect square on the left side: (v + 0.5)(v + 0.5) = 2.805555556 Calculate the square root of the right side: 1.67497927 Break this problem into two subproblems by setting (v + 0.5) equal to 1.67497927 and -1.67497927.Subproblem 1
v + 0.5 = 1.67497927 Simplifying v + 0.5 = 1.67497927 Reorder the terms: 0.5 + v = 1.67497927 Solving 0.5 + v = 1.67497927 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + v = 1.67497927 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + v = 1.67497927 + -0.5 v = 1.67497927 + -0.5 Combine like terms: 1.67497927 + -0.5 = 1.17497927 v = 1.17497927 Simplifying v = 1.17497927Subproblem 2
v + 0.5 = -1.67497927 Simplifying v + 0.5 = -1.67497927 Reorder the terms: 0.5 + v = -1.67497927 Solving 0.5 + v = -1.67497927 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + v = -1.67497927 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + v = -1.67497927 + -0.5 v = -1.67497927 + -0.5 Combine like terms: -1.67497927 + -0.5 = -2.17497927 v = -2.17497927 Simplifying v = -2.17497927Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.17497927, -2.17497927}
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