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Simplifying 8(v + 7) + -2v = 5v(3v + -4) Reorder the terms: 8(7 + v) + -2v = 5v(3v + -4) (7 * 8 + v * 8) + -2v = 5v(3v + -4) (56 + 8v) + -2v = 5v(3v + -4) Combine like terms: 8v + -2v = 6v 56 + 6v = 5v(3v + -4) Reorder the terms: 56 + 6v = 5v(-4 + 3v) 56 + 6v = (-4 * 5v + 3v * 5v) 56 + 6v = (-20v + 15v2) Solving 56 + 6v = -20v + 15v2 Solving for variable 'v'. Combine like terms: 6v + 20v = 26v 56 + 26v + -15v2 = -20v + 15v2 + 20v + -15v2 Reorder the terms: 56 + 26v + -15v2 = -20v + 20v + 15v2 + -15v2 Combine like terms: -20v + 20v = 0 56 + 26v + -15v2 = 0 + 15v2 + -15v2 56 + 26v + -15v2 = 15v2 + -15v2 Combine like terms: 15v2 + -15v2 = 0 56 + 26v + -15v2 = 0 Begin completing the square. Divide all terms by -15 the coefficient of the squared term: Divide each side by '-15'. -3.733333333 + -1.733333333v + v2 = 0 Move the constant term to the right: Add '3.733333333' to each side of the equation. -3.733333333 + -1.733333333v + 3.733333333 + v2 = 0 + 3.733333333 Reorder the terms: -3.733333333 + 3.733333333 + -1.733333333v + v2 = 0 + 3.733333333 Combine like terms: -3.733333333 + 3.733333333 = 0.000000000 0.000000000 + -1.733333333v + v2 = 0 + 3.733333333 -1.733333333v + v2 = 0 + 3.733333333 Combine like terms: 0 + 3.733333333 = 3.733333333 -1.733333333v + v2 = 3.733333333 The v term is -1.733333333v. Take half its coefficient (-0.8666666665). Square it (0.7511111108) and add it to both sides. Add '0.7511111108' to each side of the equation. -1.733333333v + 0.7511111108 + v2 = 3.733333333 + 0.7511111108 Reorder the terms: 0.7511111108 + -1.733333333v + v2 = 3.733333333 + 0.7511111108 Combine like terms: 3.733333333 + 0.7511111108 = 4.4844444438 0.7511111108 + -1.733333333v + v2 = 4.4844444438 Factor a perfect square on the left side: (v + -0.8666666665)(v + -0.8666666665) = 4.4844444438 Calculate the square root of the right side: 2.11765069 Break this problem into two subproblems by setting (v + -0.8666666665) equal to 2.11765069 and -2.11765069.Subproblem 1
v + -0.8666666665 = 2.11765069 Simplifying v + -0.8666666665 = 2.11765069 Reorder the terms: -0.8666666665 + v = 2.11765069 Solving -0.8666666665 + v = 2.11765069 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.8666666665' to each side of the equation. -0.8666666665 + 0.8666666665 + v = 2.11765069 + 0.8666666665 Combine like terms: -0.8666666665 + 0.8666666665 = 0.0000000000 0.0000000000 + v = 2.11765069 + 0.8666666665 v = 2.11765069 + 0.8666666665 Combine like terms: 2.11765069 + 0.8666666665 = 2.9843173565 v = 2.9843173565 Simplifying v = 2.9843173565Subproblem 2
v + -0.8666666665 = -2.11765069 Simplifying v + -0.8666666665 = -2.11765069 Reorder the terms: -0.8666666665 + v = -2.11765069 Solving -0.8666666665 + v = -2.11765069 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.8666666665' to each side of the equation. -0.8666666665 + 0.8666666665 + v = -2.11765069 + 0.8666666665 Combine like terms: -0.8666666665 + 0.8666666665 = 0.0000000000 0.0000000000 + v = -2.11765069 + 0.8666666665 v = -2.11765069 + 0.8666666665 Combine like terms: -2.11765069 + 0.8666666665 = -1.2509840235 v = -1.2509840235 Simplifying v = -1.2509840235Solution
The solution to the problem is based on the solutions from the subproblems. v = {2.9843173565, -1.2509840235}
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