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Simplifying 4v2 + -12v2 + -5v + 15 = 0 Reorder the terms: 15 + -5v + 4v2 + -12v2 = 0 Combine like terms: 4v2 + -12v2 = -8v2 15 + -5v + -8v2 = 0 Solving 15 + -5v + -8v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1.875 + 0.625v + v2 = 0 Move the constant term to the right: Add '1.875' to each side of the equation. -1.875 + 0.625v + 1.875 + v2 = 0 + 1.875 Reorder the terms: -1.875 + 1.875 + 0.625v + v2 = 0 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + 0.625v + v2 = 0 + 1.875 0.625v + v2 = 0 + 1.875 Combine like terms: 0 + 1.875 = 1.875 0.625v + v2 = 1.875 The v term is 0.625v. Take half its coefficient (0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. 0.625v + 0.09765625 + v2 = 1.875 + 0.09765625 Reorder the terms: 0.09765625 + 0.625v + v2 = 1.875 + 0.09765625 Combine like terms: 1.875 + 0.09765625 = 1.97265625 0.09765625 + 0.625v + v2 = 1.97265625 Factor a perfect square on the left side: (v + 0.3125)(v + 0.3125) = 1.97265625 Calculate the square root of the right side: 1.404512816 Break this problem into two subproblems by setting (v + 0.3125) equal to 1.404512816 and -1.404512816.Subproblem 1
v + 0.3125 = 1.404512816 Simplifying v + 0.3125 = 1.404512816 Reorder the terms: 0.3125 + v = 1.404512816 Solving 0.3125 + v = 1.404512816 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + v = 1.404512816 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + v = 1.404512816 + -0.3125 v = 1.404512816 + -0.3125 Combine like terms: 1.404512816 + -0.3125 = 1.092012816 v = 1.092012816 Simplifying v = 1.092012816Subproblem 2
v + 0.3125 = -1.404512816 Simplifying v + 0.3125 = -1.404512816 Reorder the terms: 0.3125 + v = -1.404512816 Solving 0.3125 + v = -1.404512816 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + v = -1.404512816 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + v = -1.404512816 + -0.3125 v = -1.404512816 + -0.3125 Combine like terms: -1.404512816 + -0.3125 = -1.717012816 v = -1.717012816 Simplifying v = -1.717012816Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.092012816, -1.717012816}
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