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Simplifying v2 + 4v + -78 = -7 Reorder the terms: -78 + 4v + v2 = -7 Solving -78 + 4v + v2 = -7 Solving for variable 'v'. Reorder the terms: -78 + 7 + 4v + v2 = -7 + 7 Combine like terms: -78 + 7 = -71 -71 + 4v + v2 = -7 + 7 Combine like terms: -7 + 7 = 0 -71 + 4v + v2 = 0 Begin completing the square. Move the constant term to the right: Add '71' to each side of the equation. -71 + 4v + 71 + v2 = 0 + 71 Reorder the terms: -71 + 71 + 4v + v2 = 0 + 71 Combine like terms: -71 + 71 = 0 0 + 4v + v2 = 0 + 71 4v + v2 = 0 + 71 Combine like terms: 0 + 71 = 71 4v + v2 = 71 The v term is 4v. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4v + 4 + v2 = 71 + 4 Reorder the terms: 4 + 4v + v2 = 71 + 4 Combine like terms: 71 + 4 = 75 4 + 4v + v2 = 75 Factor a perfect square on the left side: (v + 2)(v + 2) = 75 Calculate the square root of the right side: 8.660254038 Break this problem into two subproblems by setting (v + 2) equal to 8.660254038 and -8.660254038.Subproblem 1
v + 2 = 8.660254038 Simplifying v + 2 = 8.660254038 Reorder the terms: 2 + v = 8.660254038 Solving 2 + v = 8.660254038 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + v = 8.660254038 + -2 Combine like terms: 2 + -2 = 0 0 + v = 8.660254038 + -2 v = 8.660254038 + -2 Combine like terms: 8.660254038 + -2 = 6.660254038 v = 6.660254038 Simplifying v = 6.660254038Subproblem 2
v + 2 = -8.660254038 Simplifying v + 2 = -8.660254038 Reorder the terms: 2 + v = -8.660254038 Solving 2 + v = -8.660254038 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + v = -8.660254038 + -2 Combine like terms: 2 + -2 = 0 0 + v = -8.660254038 + -2 v = -8.660254038 + -2 Combine like terms: -8.660254038 + -2 = -10.660254038 v = -10.660254038 Simplifying v = -10.660254038Solution
The solution to the problem is based on the solutions from the subproblems. v = {6.660254038, -10.660254038}
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