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Simplifying (0.7v + -0.3)(0.11v + 1) = 0 Reorder the terms: (-0.3 + 0.7v)(0.11v + 1) = 0 Reorder the terms: (-0.3 + 0.7v)(1 + 0.11v) = 0 Multiply (-0.3 + 0.7v) * (1 + 0.11v) (-0.3(1 + 0.11v) + 0.7v * (1 + 0.11v)) = 0 ((1 * -0.3 + 0.11v * -0.3) + 0.7v * (1 + 0.11v)) = 0 ((-0.3 + -0.033v) + 0.7v * (1 + 0.11v)) = 0 (-0.3 + -0.033v + (1 * 0.7v + 0.11v * 0.7v)) = 0 (-0.3 + -0.033v + (0.7v + 0.077v2)) = 0 Combine like terms: -0.033v + 0.7v = 0.667v (-0.3 + 0.667v + 0.077v2) = 0 Solving -0.3 + 0.667v + 0.077v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 0.077 the coefficient of the squared term: Divide each side by '0.077'. -3.896103896 + 8.662337662v + v2 = 0 Move the constant term to the right: Add '3.896103896' to each side of the equation. -3.896103896 + 8.662337662v + 3.896103896 + v2 = 0 + 3.896103896 Reorder the terms: -3.896103896 + 3.896103896 + 8.662337662v + v2 = 0 + 3.896103896 Combine like terms: -3.896103896 + 3.896103896 = 0.000000000 0.000000000 + 8.662337662v + v2 = 0 + 3.896103896 8.662337662v + v2 = 0 + 3.896103896 Combine like terms: 0 + 3.896103896 = 3.896103896 8.662337662v + v2 = 3.896103896 The v term is 8.662337662v. Take half its coefficient (4.331168831). Square it (18.75902344) and add it to both sides. Add '18.75902344' to each side of the equation. 8.662337662v + 18.75902344 + v2 = 3.896103896 + 18.75902344 Reorder the terms: 18.75902344 + 8.662337662v + v2 = 3.896103896 + 18.75902344 Combine like terms: 3.896103896 + 18.75902344 = 22.655127336 18.75902344 + 8.662337662v + v2 = 22.655127336 Factor a perfect square on the left side: (v + 4.331168831)(v + 4.331168831) = 22.655127336 Calculate the square root of the right side: 4.759740259 Break this problem into two subproblems by setting (v + 4.331168831) equal to 4.759740259 and -4.759740259.Subproblem 1
v + 4.331168831 = 4.759740259 Simplifying v + 4.331168831 = 4.759740259 Reorder the terms: 4.331168831 + v = 4.759740259 Solving 4.331168831 + v = 4.759740259 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-4.331168831' to each side of the equation. 4.331168831 + -4.331168831 + v = 4.759740259 + -4.331168831 Combine like terms: 4.331168831 + -4.331168831 = 0.000000000 0.000000000 + v = 4.759740259 + -4.331168831 v = 4.759740259 + -4.331168831 Combine like terms: 4.759740259 + -4.331168831 = 0.428571428 v = 0.428571428 Simplifying v = 0.428571428Subproblem 2
v + 4.331168831 = -4.759740259 Simplifying v + 4.331168831 = -4.759740259 Reorder the terms: 4.331168831 + v = -4.759740259 Solving 4.331168831 + v = -4.759740259 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-4.331168831' to each side of the equation. 4.331168831 + -4.331168831 + v = -4.759740259 + -4.331168831 Combine like terms: 4.331168831 + -4.331168831 = 0.000000000 0.000000000 + v = -4.759740259 + -4.331168831 v = -4.759740259 + -4.331168831 Combine like terms: -4.759740259 + -4.331168831 = -9.09090909 v = -9.09090909 Simplifying v = -9.09090909Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.428571428, -9.09090909}
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