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Simplifying 2v2 + 12v + 25 = v2 + 1 Reorder the terms: 25 + 12v + 2v2 = v2 + 1 Reorder the terms: 25 + 12v + 2v2 = 1 + v2 Solving 25 + 12v + 2v2 = 1 + v2 Solving for variable 'v'. Reorder the terms: 25 + -1 + 12v + 2v2 + -1v2 = 1 + v2 + -1 + -1v2 Combine like terms: 25 + -1 = 24 24 + 12v + 2v2 + -1v2 = 1 + v2 + -1 + -1v2 Combine like terms: 2v2 + -1v2 = 1v2 24 + 12v + 1v2 = 1 + v2 + -1 + -1v2 Reorder the terms: 24 + 12v + 1v2 = 1 + -1 + v2 + -1v2 Combine like terms: 1 + -1 = 0 24 + 12v + 1v2 = 0 + v2 + -1v2 24 + 12v + 1v2 = v2 + -1v2 Combine like terms: v2 + -1v2 = 0 24 + 12v + 1v2 = 0 Begin completing the square. Move the constant term to the right: Add '-24' to each side of the equation. 24 + 12v + -24 + v2 = 0 + -24 Reorder the terms: 24 + -24 + 12v + v2 = 0 + -24 Combine like terms: 24 + -24 = 0 0 + 12v + v2 = 0 + -24 12v + v2 = 0 + -24 Combine like terms: 0 + -24 = -24 12v + v2 = -24 The v term is 12v. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12v + 36 + v2 = -24 + 36 Reorder the terms: 36 + 12v + v2 = -24 + 36 Combine like terms: -24 + 36 = 12 36 + 12v + v2 = 12 Factor a perfect square on the left side: (v + 6)(v + 6) = 12 Calculate the square root of the right side: 3.464101615 Break this problem into two subproblems by setting (v + 6) equal to 3.464101615 and -3.464101615.Subproblem 1
v + 6 = 3.464101615 Simplifying v + 6 = 3.464101615 Reorder the terms: 6 + v = 3.464101615 Solving 6 + v = 3.464101615 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + v = 3.464101615 + -6 Combine like terms: 6 + -6 = 0 0 + v = 3.464101615 + -6 v = 3.464101615 + -6 Combine like terms: 3.464101615 + -6 = -2.535898385 v = -2.535898385 Simplifying v = -2.535898385Subproblem 2
v + 6 = -3.464101615 Simplifying v + 6 = -3.464101615 Reorder the terms: 6 + v = -3.464101615 Solving 6 + v = -3.464101615 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + v = -3.464101615 + -6 Combine like terms: 6 + -6 = 0 0 + v = -3.464101615 + -6 v = -3.464101615 + -6 Combine like terms: -3.464101615 + -6 = -9.464101615 v = -9.464101615 Simplifying v = -9.464101615Solution
The solution to the problem is based on the solutions from the subproblems. v = {-2.535898385, -9.464101615}
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