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Simplifying -8(v + 4) * 4v + 7 = 5v + 8 Reorder the terms: -8(4 + v) * 4v + 7 = 5v + 8 Reorder the terms for easier multiplication: -8 * 4v(4 + v) + 7 = 5v + 8 Multiply -8 * 4 -32v(4 + v) + 7 = 5v + 8 (4 * -32v + v * -32v) + 7 = 5v + 8 (-128v + -32v2) + 7 = 5v + 8 Reorder the terms: 7 + -128v + -32v2 = 5v + 8 Reorder the terms: 7 + -128v + -32v2 = 8 + 5v Solving 7 + -128v + -32v2 = 8 + 5v Solving for variable 'v'. Reorder the terms: 7 + -8 + -128v + -5v + -32v2 = 8 + 5v + -8 + -5v Combine like terms: 7 + -8 = -1 -1 + -128v + -5v + -32v2 = 8 + 5v + -8 + -5v Combine like terms: -128v + -5v = -133v -1 + -133v + -32v2 = 8 + 5v + -8 + -5v Reorder the terms: -1 + -133v + -32v2 = 8 + -8 + 5v + -5v Combine like terms: 8 + -8 = 0 -1 + -133v + -32v2 = 0 + 5v + -5v -1 + -133v + -32v2 = 5v + -5v Combine like terms: 5v + -5v = 0 -1 + -133v + -32v2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(1 + 133v + 32v2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(1 + 133v + 32v2)' equal to zero and attempt to solve: Simplifying 1 + 133v + 32v2 = 0 Solving 1 + 133v + 32v2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. 0.03125 + 4.15625v + v2 = 0 Move the constant term to the right: Add '-0.03125' to each side of the equation. 0.03125 + 4.15625v + -0.03125 + v2 = 0 + -0.03125 Reorder the terms: 0.03125 + -0.03125 + 4.15625v + v2 = 0 + -0.03125 Combine like terms: 0.03125 + -0.03125 = 0.00000 0.00000 + 4.15625v + v2 = 0 + -0.03125 4.15625v + v2 = 0 + -0.03125 Combine like terms: 0 + -0.03125 = -0.03125 4.15625v + v2 = -0.03125 The v term is 4.15625v. Take half its coefficient (2.078125). Square it (4.318603516) and add it to both sides. Add '4.318603516' to each side of the equation. 4.15625v + 4.318603516 + v2 = -0.03125 + 4.318603516 Reorder the terms: 4.318603516 + 4.15625v + v2 = -0.03125 + 4.318603516 Combine like terms: -0.03125 + 4.318603516 = 4.287353516 4.318603516 + 4.15625v + v2 = 4.287353516 Factor a perfect square on the left side: (v + 2.078125)(v + 2.078125) = 4.287353516 Calculate the square root of the right side: 2.070592552 Break this problem into two subproblems by setting (v + 2.078125) equal to 2.070592552 and -2.070592552.Subproblem 1
v + 2.078125 = 2.070592552 Simplifying v + 2.078125 = 2.070592552 Reorder the terms: 2.078125 + v = 2.070592552 Solving 2.078125 + v = 2.070592552 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2.078125' to each side of the equation. 2.078125 + -2.078125 + v = 2.070592552 + -2.078125 Combine like terms: 2.078125 + -2.078125 = 0.000000 0.000000 + v = 2.070592552 + -2.078125 v = 2.070592552 + -2.078125 Combine like terms: 2.070592552 + -2.078125 = -0.007532448 v = -0.007532448 Simplifying v = -0.007532448Subproblem 2
v + 2.078125 = -2.070592552 Simplifying v + 2.078125 = -2.070592552 Reorder the terms: 2.078125 + v = -2.070592552 Solving 2.078125 + v = -2.070592552 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2.078125' to each side of the equation. 2.078125 + -2.078125 + v = -2.070592552 + -2.078125 Combine like terms: 2.078125 + -2.078125 = 0.000000 0.000000 + v = -2.070592552 + -2.078125 v = -2.070592552 + -2.078125 Combine like terms: -2.070592552 + -2.078125 = -4.148717552 v = -4.148717552 Simplifying v = -4.148717552Solution
The solution to the problem is based on the solutions from the subproblems. v = {-0.007532448, -4.148717552}Solution
v = {-0.007532448, -4.148717552}
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