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Simplifying v(3v + -23) = -21 Reorder the terms: v(-23 + 3v) = -21 (-23 * v + 3v * v) = -21 (-23v + 3v2) = -21 Solving -23v + 3v2 = -21 Solving for variable 'v'. Reorder the terms: 21 + -23v + 3v2 = -21 + 21 Combine like terms: -21 + 21 = 0 21 + -23v + 3v2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 7 + -7.666666667v + v2 = 0 Move the constant term to the right: Add '-7' to each side of the equation. 7 + -7.666666667v + -7 + v2 = 0 + -7 Reorder the terms: 7 + -7 + -7.666666667v + v2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + -7.666666667v + v2 = 0 + -7 -7.666666667v + v2 = 0 + -7 Combine like terms: 0 + -7 = -7 -7.666666667v + v2 = -7 The v term is -7.666666667v. Take half its coefficient (-3.833333334). Square it (14.69444445) and add it to both sides. Add '14.69444445' to each side of the equation. -7.666666667v + 14.69444445 + v2 = -7 + 14.69444445 Reorder the terms: 14.69444445 + -7.666666667v + v2 = -7 + 14.69444445 Combine like terms: -7 + 14.69444445 = 7.69444445 14.69444445 + -7.666666667v + v2 = 7.69444445 Factor a perfect square on the left side: (v + -3.833333334)(v + -3.833333334) = 7.69444445 Calculate the square root of the right side: 2.773886164 Break this problem into two subproblems by setting (v + -3.833333334) equal to 2.773886164 and -2.773886164.Subproblem 1
v + -3.833333334 = 2.773886164 Simplifying v + -3.833333334 = 2.773886164 Reorder the terms: -3.833333334 + v = 2.773886164 Solving -3.833333334 + v = 2.773886164 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '3.833333334' to each side of the equation. -3.833333334 + 3.833333334 + v = 2.773886164 + 3.833333334 Combine like terms: -3.833333334 + 3.833333334 = 0.000000000 0.000000000 + v = 2.773886164 + 3.833333334 v = 2.773886164 + 3.833333334 Combine like terms: 2.773886164 + 3.833333334 = 6.607219498 v = 6.607219498 Simplifying v = 6.607219498Subproblem 2
v + -3.833333334 = -2.773886164 Simplifying v + -3.833333334 = -2.773886164 Reorder the terms: -3.833333334 + v = -2.773886164 Solving -3.833333334 + v = -2.773886164 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '3.833333334' to each side of the equation. -3.833333334 + 3.833333334 + v = -2.773886164 + 3.833333334 Combine like terms: -3.833333334 + 3.833333334 = 0.000000000 0.000000000 + v = -2.773886164 + 3.833333334 v = -2.773886164 + 3.833333334 Combine like terms: -2.773886164 + 3.833333334 = 1.05944717 v = 1.05944717 Simplifying v = 1.05944717Solution
The solution to the problem is based on the solutions from the subproblems. v = {6.607219498, 1.05944717}
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