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Simplifying 2v2 + 13v + 24 = v2 + 36 Reorder the terms: 24 + 13v + 2v2 = v2 + 36 Reorder the terms: 24 + 13v + 2v2 = 36 + v2 Solving 24 + 13v + 2v2 = 36 + v2 Solving for variable 'v'. Reorder the terms: 24 + -36 + 13v + 2v2 + -1v2 = 36 + v2 + -36 + -1v2 Combine like terms: 24 + -36 = -12 -12 + 13v + 2v2 + -1v2 = 36 + v2 + -36 + -1v2 Combine like terms: 2v2 + -1v2 = 1v2 -12 + 13v + 1v2 = 36 + v2 + -36 + -1v2 Reorder the terms: -12 + 13v + 1v2 = 36 + -36 + v2 + -1v2 Combine like terms: 36 + -36 = 0 -12 + 13v + 1v2 = 0 + v2 + -1v2 -12 + 13v + 1v2 = v2 + -1v2 Combine like terms: v2 + -1v2 = 0 -12 + 13v + 1v2 = 0 Begin completing the square. Move the constant term to the right: Add '12' to each side of the equation. -12 + 13v + 12 + v2 = 0 + 12 Reorder the terms: -12 + 12 + 13v + v2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 13v + v2 = 0 + 12 13v + v2 = 0 + 12 Combine like terms: 0 + 12 = 12 13v + v2 = 12 The v term is 13v. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13v + 42.25 + v2 = 12 + 42.25 Reorder the terms: 42.25 + 13v + v2 = 12 + 42.25 Combine like terms: 12 + 42.25 = 54.25 42.25 + 13v + v2 = 54.25 Factor a perfect square on the left side: (v + 6.5)(v + 6.5) = 54.25 Calculate the square root of the right side: 7.365459931 Break this problem into two subproblems by setting (v + 6.5) equal to 7.365459931 and -7.365459931.Subproblem 1
v + 6.5 = 7.365459931 Simplifying v + 6.5 = 7.365459931 Reorder the terms: 6.5 + v = 7.365459931 Solving 6.5 + v = 7.365459931 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + v = 7.365459931 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + v = 7.365459931 + -6.5 v = 7.365459931 + -6.5 Combine like terms: 7.365459931 + -6.5 = 0.865459931 v = 0.865459931 Simplifying v = 0.865459931Subproblem 2
v + 6.5 = -7.365459931 Simplifying v + 6.5 = -7.365459931 Reorder the terms: 6.5 + v = -7.365459931 Solving 6.5 + v = -7.365459931 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + v = -7.365459931 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + v = -7.365459931 + -6.5 v = -7.365459931 + -6.5 Combine like terms: -7.365459931 + -6.5 = -13.865459931 v = -13.865459931 Simplifying v = -13.865459931Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.865459931, -13.865459931}
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