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Simplifying 4v(v + -2) = 7 + -2(3v + -4) Reorder the terms: 4v(-2 + v) = 7 + -2(3v + -4) (-2 * 4v + v * 4v) = 7 + -2(3v + -4) (-8v + 4v2) = 7 + -2(3v + -4) Reorder the terms: -8v + 4v2 = 7 + -2(-4 + 3v) -8v + 4v2 = 7 + (-4 * -2 + 3v * -2) -8v + 4v2 = 7 + (8 + -6v) Combine like terms: 7 + 8 = 15 -8v + 4v2 = 15 + -6v Solving -8v + 4v2 = 15 + -6v Solving for variable 'v'. Reorder the terms: -15 + -8v + 6v + 4v2 = 15 + -6v + -15 + 6v Combine like terms: -8v + 6v = -2v -15 + -2v + 4v2 = 15 + -6v + -15 + 6v Reorder the terms: -15 + -2v + 4v2 = 15 + -15 + -6v + 6v Combine like terms: 15 + -15 = 0 -15 + -2v + 4v2 = 0 + -6v + 6v -15 + -2v + 4v2 = -6v + 6v Combine like terms: -6v + 6v = 0 -15 + -2v + 4v2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -3.75 + -0.5v + v2 = 0 Move the constant term to the right: Add '3.75' to each side of the equation. -3.75 + -0.5v + 3.75 + v2 = 0 + 3.75 Reorder the terms: -3.75 + 3.75 + -0.5v + v2 = 0 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + -0.5v + v2 = 0 + 3.75 -0.5v + v2 = 0 + 3.75 Combine like terms: 0 + 3.75 = 3.75 -0.5v + v2 = 3.75 The v term is -0.5v. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5v + 0.0625 + v2 = 3.75 + 0.0625 Reorder the terms: 0.0625 + -0.5v + v2 = 3.75 + 0.0625 Combine like terms: 3.75 + 0.0625 = 3.8125 0.0625 + -0.5v + v2 = 3.8125 Factor a perfect square on the left side: (v + -0.25)(v + -0.25) = 3.8125 Calculate the square root of the right side: 1.952562419 Break this problem into two subproblems by setting (v + -0.25) equal to 1.952562419 and -1.952562419.Subproblem 1
v + -0.25 = 1.952562419 Simplifying v + -0.25 = 1.952562419 Reorder the terms: -0.25 + v = 1.952562419 Solving -0.25 + v = 1.952562419 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + v = 1.952562419 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + v = 1.952562419 + 0.25 v = 1.952562419 + 0.25 Combine like terms: 1.952562419 + 0.25 = 2.202562419 v = 2.202562419 Simplifying v = 2.202562419Subproblem 2
v + -0.25 = -1.952562419 Simplifying v + -0.25 = -1.952562419 Reorder the terms: -0.25 + v = -1.952562419 Solving -0.25 + v = -1.952562419 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + v = -1.952562419 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + v = -1.952562419 + 0.25 v = -1.952562419 + 0.25 Combine like terms: -1.952562419 + 0.25 = -1.702562419 v = -1.702562419 Simplifying v = -1.702562419Solution
The solution to the problem is based on the solutions from the subproblems. v = {2.202562419, -1.702562419}
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