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Simplifying v(v + -29.7) + (v + 23.9) = 194.3 Reorder the terms: v(-29.7 + v) + (v + 23.9) = 194.3 (-29.7 * v + v * v) + (v + 23.9) = 194.3 (-29.7v + v2) + (v + 23.9) = 194.3 Reorder the terms: -29.7v + v2 + (23.9 + v) = 194.3 Remove parenthesis around (23.9 + v) -29.7v + v2 + 23.9 + v = 194.3 Reorder the terms: 23.9 + -29.7v + v + v2 = 194.3 Combine like terms: -29.7v + v = -28.7v 23.9 + -28.7v + v2 = 194.3 Solving 23.9 + -28.7v + v2 = 194.3 Solving for variable 'v'. Reorder the terms: 23.9 + -194.3 + -28.7v + v2 = 194.3 + -194.3 Combine like terms: 23.9 + -194.3 = -170.4 -170.4 + -28.7v + v2 = 194.3 + -194.3 Combine like terms: 194.3 + -194.3 = 0.0 -170.4 + -28.7v + v2 = 0.0 Begin completing the square. Move the constant term to the right: Add '170.4' to each side of the equation. -170.4 + -28.7v + 170.4 + v2 = 0.0 + 170.4 Reorder the terms: -170.4 + 170.4 + -28.7v + v2 = 0.0 + 170.4 Combine like terms: -170.4 + 170.4 = 0.0 0.0 + -28.7v + v2 = 0.0 + 170.4 -28.7v + v2 = 0.0 + 170.4 Combine like terms: 0.0 + 170.4 = 170.4 -28.7v + v2 = 170.4 The v term is -28.7v. Take half its coefficient (-14.35). Square it (205.9225) and add it to both sides. Add '205.9225' to each side of the equation. -28.7v + 205.9225 + v2 = 170.4 + 205.9225 Reorder the terms: 205.9225 + -28.7v + v2 = 170.4 + 205.9225 Combine like terms: 170.4 + 205.9225 = 376.3225 205.9225 + -28.7v + v2 = 376.3225 Factor a perfect square on the left side: (v + -14.35)(v + -14.35) = 376.3225 Calculate the square root of the right side: 19.399033481 Break this problem into two subproblems by setting (v + -14.35) equal to 19.399033481 and -19.399033481.Subproblem 1
v + -14.35 = 19.399033481 Simplifying v + -14.35 = 19.399033481 Reorder the terms: -14.35 + v = 19.399033481 Solving -14.35 + v = 19.399033481 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '14.35' to each side of the equation. -14.35 + 14.35 + v = 19.399033481 + 14.35 Combine like terms: -14.35 + 14.35 = 0.00 0.00 + v = 19.399033481 + 14.35 v = 19.399033481 + 14.35 Combine like terms: 19.399033481 + 14.35 = 33.749033481 v = 33.749033481 Simplifying v = 33.749033481Subproblem 2
v + -14.35 = -19.399033481 Simplifying v + -14.35 = -19.399033481 Reorder the terms: -14.35 + v = -19.399033481 Solving -14.35 + v = -19.399033481 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '14.35' to each side of the equation. -14.35 + 14.35 + v = -19.399033481 + 14.35 Combine like terms: -14.35 + 14.35 = 0.00 0.00 + v = -19.399033481 + 14.35 v = -19.399033481 + 14.35 Combine like terms: -19.399033481 + 14.35 = -5.049033481 v = -5.049033481 Simplifying v = -5.049033481Solution
The solution to the problem is based on the solutions from the subproblems. v = {33.749033481, -5.049033481}
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