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Simplifying (v + -8)(v + -8) + -32 = 0 Reorder the terms: (-8 + v)(v + -8) + -32 = 0 Reorder the terms: (-8 + v)(-8 + v) + -32 = 0 Multiply (-8 + v) * (-8 + v) (-8(-8 + v) + v(-8 + v)) + -32 = 0 ((-8 * -8 + v * -8) + v(-8 + v)) + -32 = 0 ((64 + -8v) + v(-8 + v)) + -32 = 0 (64 + -8v + (-8 * v + v * v)) + -32 = 0 (64 + -8v + (-8v + v2)) + -32 = 0 Combine like terms: -8v + -8v = -16v (64 + -16v + v2) + -32 = 0 Reorder the terms: 64 + -32 + -16v + v2 = 0 Combine like terms: 64 + -32 = 32 32 + -16v + v2 = 0 Solving 32 + -16v + v2 = 0 Solving for variable 'v'. Begin completing the square. Move the constant term to the right: Add '-32' to each side of the equation. 32 + -16v + -32 + v2 = 0 + -32 Reorder the terms: 32 + -32 + -16v + v2 = 0 + -32 Combine like terms: 32 + -32 = 0 0 + -16v + v2 = 0 + -32 -16v + v2 = 0 + -32 Combine like terms: 0 + -32 = -32 -16v + v2 = -32 The v term is -16v. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16v + 64 + v2 = -32 + 64 Reorder the terms: 64 + -16v + v2 = -32 + 64 Combine like terms: -32 + 64 = 32 64 + -16v + v2 = 32 Factor a perfect square on the left side: (v + -8)(v + -8) = 32 Calculate the square root of the right side: 5.656854249 Break this problem into two subproblems by setting (v + -8) equal to 5.656854249 and -5.656854249.Subproblem 1
v + -8 = 5.656854249 Simplifying v + -8 = 5.656854249 Reorder the terms: -8 + v = 5.656854249 Solving -8 + v = 5.656854249 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + v = 5.656854249 + 8 Combine like terms: -8 + 8 = 0 0 + v = 5.656854249 + 8 v = 5.656854249 + 8 Combine like terms: 5.656854249 + 8 = 13.656854249 v = 13.656854249 Simplifying v = 13.656854249Subproblem 2
v + -8 = -5.656854249 Simplifying v + -8 = -5.656854249 Reorder the terms: -8 + v = -5.656854249 Solving -8 + v = -5.656854249 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + v = -5.656854249 + 8 Combine like terms: -8 + 8 = 0 0 + v = -5.656854249 + 8 v = -5.656854249 + 8 Combine like terms: -5.656854249 + 8 = 2.343145751 v = 2.343145751 Simplifying v = 2.343145751Solution
The solution to the problem is based on the solutions from the subproblems. v = {13.656854249, 2.343145751}
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