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Simplifying v(4v + -23) = -21 Reorder the terms: v(-23 + 4v) = -21 (-23 * v + 4v * v) = -21 (-23v + 4v2) = -21 Solving -23v + 4v2 = -21 Solving for variable 'v'. Reorder the terms: 21 + -23v + 4v2 = -21 + 21 Combine like terms: -21 + 21 = 0 21 + -23v + 4v2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 5.25 + -5.75v + v2 = 0 Move the constant term to the right: Add '-5.25' to each side of the equation. 5.25 + -5.75v + -5.25 + v2 = 0 + -5.25 Reorder the terms: 5.25 + -5.25 + -5.75v + v2 = 0 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + -5.75v + v2 = 0 + -5.25 -5.75v + v2 = 0 + -5.25 Combine like terms: 0 + -5.25 = -5.25 -5.75v + v2 = -5.25 The v term is -5.75v. Take half its coefficient (-2.875). Square it (8.265625) and add it to both sides. Add '8.265625' to each side of the equation. -5.75v + 8.265625 + v2 = -5.25 + 8.265625 Reorder the terms: 8.265625 + -5.75v + v2 = -5.25 + 8.265625 Combine like terms: -5.25 + 8.265625 = 3.015625 8.265625 + -5.75v + v2 = 3.015625 Factor a perfect square on the left side: (v + -2.875)(v + -2.875) = 3.015625 Calculate the square root of the right side: 1.736555499 Break this problem into two subproblems by setting (v + -2.875) equal to 1.736555499 and -1.736555499.Subproblem 1
v + -2.875 = 1.736555499 Simplifying v + -2.875 = 1.736555499 Reorder the terms: -2.875 + v = 1.736555499 Solving -2.875 + v = 1.736555499 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '2.875' to each side of the equation. -2.875 + 2.875 + v = 1.736555499 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + v = 1.736555499 + 2.875 v = 1.736555499 + 2.875 Combine like terms: 1.736555499 + 2.875 = 4.611555499 v = 4.611555499 Simplifying v = 4.611555499Subproblem 2
v + -2.875 = -1.736555499 Simplifying v + -2.875 = -1.736555499 Reorder the terms: -2.875 + v = -1.736555499 Solving -2.875 + v = -1.736555499 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '2.875' to each side of the equation. -2.875 + 2.875 + v = -1.736555499 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + v = -1.736555499 + 2.875 v = -1.736555499 + 2.875 Combine like terms: -1.736555499 + 2.875 = 1.138444501 v = 1.138444501 Simplifying v = 1.138444501Solution
The solution to the problem is based on the solutions from the subproblems. v = {4.611555499, 1.138444501}
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