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Simplifying 2(v + 8) = -4(4v + -8) * 8v Reorder the terms: 2(8 + v) = -4(4v + -8) * 8v (8 * 2 + v * 2) = -4(4v + -8) * 8v (16 + 2v) = -4(4v + -8) * 8v Reorder the terms: 16 + 2v = -4(-8 + 4v) * 8v Reorder the terms for easier multiplication: 16 + 2v = -4 * 8v(-8 + 4v) Multiply -4 * 8 16 + 2v = -32v(-8 + 4v) 16 + 2v = (-8 * -32v + 4v * -32v) 16 + 2v = (256v + -128v2) Solving 16 + 2v = 256v + -128v2 Solving for variable 'v'. Combine like terms: 2v + -256v = -254v 16 + -254v + 128v2 = 256v + -128v2 + -256v + 128v2 Reorder the terms: 16 + -254v + 128v2 = 256v + -256v + -128v2 + 128v2 Combine like terms: 256v + -256v = 0 16 + -254v + 128v2 = 0 + -128v2 + 128v2 16 + -254v + 128v2 = -128v2 + 128v2 Combine like terms: -128v2 + 128v2 = 0 16 + -254v + 128v2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(8 + -127v + 64v2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(8 + -127v + 64v2)' equal to zero and attempt to solve: Simplifying 8 + -127v + 64v2 = 0 Solving 8 + -127v + 64v2 = 0 Begin completing the square. Divide all terms by 64 the coefficient of the squared term: Divide each side by '64'. 0.125 + -1.984375v + v2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + -1.984375v + -0.125 + v2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + -1.984375v + v2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + -1.984375v + v2 = 0 + -0.125 -1.984375v + v2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 -1.984375v + v2 = -0.125 The v term is -1.984375v. Take half its coefficient (-0.9921875). Square it (0.9844360352) and add it to both sides. Add '0.9844360352' to each side of the equation. -1.984375v + 0.9844360352 + v2 = -0.125 + 0.9844360352 Reorder the terms: 0.9844360352 + -1.984375v + v2 = -0.125 + 0.9844360352 Combine like terms: -0.125 + 0.9844360352 = 0.8594360352 0.9844360352 + -1.984375v + v2 = 0.8594360352 Factor a perfect square on the left side: (v + -0.9921875)(v + -0.9921875) = 0.8594360352 Calculate the square root of the right side: 0.92705773 Break this problem into two subproblems by setting (v + -0.9921875) equal to 0.92705773 and -0.92705773.Subproblem 1
v + -0.9921875 = 0.92705773 Simplifying v + -0.9921875 = 0.92705773 Reorder the terms: -0.9921875 + v = 0.92705773 Solving -0.9921875 + v = 0.92705773 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.9921875' to each side of the equation. -0.9921875 + 0.9921875 + v = 0.92705773 + 0.9921875 Combine like terms: -0.9921875 + 0.9921875 = 0.0000000 0.0000000 + v = 0.92705773 + 0.9921875 v = 0.92705773 + 0.9921875 Combine like terms: 0.92705773 + 0.9921875 = 1.91924523 v = 1.91924523 Simplifying v = 1.91924523Subproblem 2
v + -0.9921875 = -0.92705773 Simplifying v + -0.9921875 = -0.92705773 Reorder the terms: -0.9921875 + v = -0.92705773 Solving -0.9921875 + v = -0.92705773 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.9921875' to each side of the equation. -0.9921875 + 0.9921875 + v = -0.92705773 + 0.9921875 Combine like terms: -0.9921875 + 0.9921875 = 0.0000000 0.0000000 + v = -0.92705773 + 0.9921875 v = -0.92705773 + 0.9921875 Combine like terms: -0.92705773 + 0.9921875 = 0.06512977 v = 0.06512977 Simplifying v = 0.06512977Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.91924523, 0.06512977}Solution
v = {1.91924523, 0.06512977}
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