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Simplifying 3(7 + 8v) + 4v(v + 7) = 0 (7 * 3 + 8v * 3) + 4v(v + 7) = 0 (21 + 24v) + 4v(v + 7) = 0 Reorder the terms: 21 + 24v + 4v(7 + v) = 0 21 + 24v + (7 * 4v + v * 4v) = 0 21 + 24v + (28v + 4v2) = 0 Combine like terms: 24v + 28v = 52v 21 + 52v + 4v2 = 0 Solving 21 + 52v + 4v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 5.25 + 13v + v2 = 0 Move the constant term to the right: Add '-5.25' to each side of the equation. 5.25 + 13v + -5.25 + v2 = 0 + -5.25 Reorder the terms: 5.25 + -5.25 + 13v + v2 = 0 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + 13v + v2 = 0 + -5.25 13v + v2 = 0 + -5.25 Combine like terms: 0 + -5.25 = -5.25 13v + v2 = -5.25 The v term is 13v. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13v + 42.25 + v2 = -5.25 + 42.25 Reorder the terms: 42.25 + 13v + v2 = -5.25 + 42.25 Combine like terms: -5.25 + 42.25 = 37 42.25 + 13v + v2 = 37 Factor a perfect square on the left side: (v + 6.5)(v + 6.5) = 37 Calculate the square root of the right side: 6.08276253 Break this problem into two subproblems by setting (v + 6.5) equal to 6.08276253 and -6.08276253.Subproblem 1
v + 6.5 = 6.08276253 Simplifying v + 6.5 = 6.08276253 Reorder the terms: 6.5 + v = 6.08276253 Solving 6.5 + v = 6.08276253 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + v = 6.08276253 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + v = 6.08276253 + -6.5 v = 6.08276253 + -6.5 Combine like terms: 6.08276253 + -6.5 = -0.41723747 v = -0.41723747 Simplifying v = -0.41723747Subproblem 2
v + 6.5 = -6.08276253 Simplifying v + 6.5 = -6.08276253 Reorder the terms: 6.5 + v = -6.08276253 Solving 6.5 + v = -6.08276253 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + v = -6.08276253 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + v = -6.08276253 + -6.5 v = -6.08276253 + -6.5 Combine like terms: -6.08276253 + -6.5 = -12.58276253 v = -12.58276253 Simplifying v = -12.58276253Solution
The solution to the problem is based on the solutions from the subproblems. v = {-0.41723747, -12.58276253}
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