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Simplifying 7 + -1v(4v + 1) = 0 Reorder the terms: 7 + -1v(1 + 4v) = 0 7 + (1 * -1v + 4v * -1v) = 0 7 + (-1v + -4v2) = 0 Solving 7 + -1v + -4v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1.75 + 0.25v + v2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + 0.25v + 1.75 + v2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + 0.25v + v2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + 0.25v + v2 = 0 + 1.75 0.25v + v2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 0.25v + v2 = 1.75 The v term is 0.25v. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25v + 0.015625 + v2 = 1.75 + 0.015625 Reorder the terms: 0.015625 + 0.25v + v2 = 1.75 + 0.015625 Combine like terms: 1.75 + 0.015625 = 1.765625 0.015625 + 0.25v + v2 = 1.765625 Factor a perfect square on the left side: (v + 0.125)(v + 0.125) = 1.765625 Calculate the square root of the right side: 1.328768227 Break this problem into two subproblems by setting (v + 0.125) equal to 1.328768227 and -1.328768227.Subproblem 1
v + 0.125 = 1.328768227 Simplifying v + 0.125 = 1.328768227 Reorder the terms: 0.125 + v = 1.328768227 Solving 0.125 + v = 1.328768227 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + v = 1.328768227 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + v = 1.328768227 + -0.125 v = 1.328768227 + -0.125 Combine like terms: 1.328768227 + -0.125 = 1.203768227 v = 1.203768227 Simplifying v = 1.203768227Subproblem 2
v + 0.125 = -1.328768227 Simplifying v + 0.125 = -1.328768227 Reorder the terms: 0.125 + v = -1.328768227 Solving 0.125 + v = -1.328768227 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + v = -1.328768227 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + v = -1.328768227 + -0.125 v = -1.328768227 + -0.125 Combine like terms: -1.328768227 + -0.125 = -1.453768227 v = -1.453768227 Simplifying v = -1.453768227Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.203768227, -1.453768227}
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