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Simplifying (16v + 3)(v + 3) = 4 Reorder the terms: (3 + 16v)(v + 3) = 4 Reorder the terms: (3 + 16v)(3 + v) = 4 Multiply (3 + 16v) * (3 + v) (3(3 + v) + 16v * (3 + v)) = 4 ((3 * 3 + v * 3) + 16v * (3 + v)) = 4 ((9 + 3v) + 16v * (3 + v)) = 4 (9 + 3v + (3 * 16v + v * 16v)) = 4 (9 + 3v + (48v + 16v2)) = 4 Combine like terms: 3v + 48v = 51v (9 + 51v + 16v2) = 4 Solving 9 + 51v + 16v2 = 4 Solving for variable 'v'. Reorder the terms: 9 + -4 + 51v + 16v2 = 4 + -4 Combine like terms: 9 + -4 = 5 5 + 51v + 16v2 = 4 + -4 Combine like terms: 4 + -4 = 0 5 + 51v + 16v2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.3125 + 3.1875v + v2 = 0 Move the constant term to the right: Add '-0.3125' to each side of the equation. 0.3125 + 3.1875v + -0.3125 + v2 = 0 + -0.3125 Reorder the terms: 0.3125 + -0.3125 + 3.1875v + v2 = 0 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + 3.1875v + v2 = 0 + -0.3125 3.1875v + v2 = 0 + -0.3125 Combine like terms: 0 + -0.3125 = -0.3125 3.1875v + v2 = -0.3125 The v term is 3.1875v. Take half its coefficient (1.59375). Square it (2.540039063) and add it to both sides. Add '2.540039063' to each side of the equation. 3.1875v + 2.540039063 + v2 = -0.3125 + 2.540039063 Reorder the terms: 2.540039063 + 3.1875v + v2 = -0.3125 + 2.540039063 Combine like terms: -0.3125 + 2.540039063 = 2.227539063 2.540039063 + 3.1875v + v2 = 2.227539063 Factor a perfect square on the left side: (v + 1.59375)(v + 1.59375) = 2.227539063 Calculate the square root of the right side: 1.492494242 Break this problem into two subproblems by setting (v + 1.59375) equal to 1.492494242 and -1.492494242.Subproblem 1
v + 1.59375 = 1.492494242 Simplifying v + 1.59375 = 1.492494242 Reorder the terms: 1.59375 + v = 1.492494242 Solving 1.59375 + v = 1.492494242 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.59375' to each side of the equation. 1.59375 + -1.59375 + v = 1.492494242 + -1.59375 Combine like terms: 1.59375 + -1.59375 = 0.00000 0.00000 + v = 1.492494242 + -1.59375 v = 1.492494242 + -1.59375 Combine like terms: 1.492494242 + -1.59375 = -0.101255758 v = -0.101255758 Simplifying v = -0.101255758Subproblem 2
v + 1.59375 = -1.492494242 Simplifying v + 1.59375 = -1.492494242 Reorder the terms: 1.59375 + v = -1.492494242 Solving 1.59375 + v = -1.492494242 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.59375' to each side of the equation. 1.59375 + -1.59375 + v = -1.492494242 + -1.59375 Combine like terms: 1.59375 + -1.59375 = 0.00000 0.00000 + v = -1.492494242 + -1.59375 v = -1.492494242 + -1.59375 Combine like terms: -1.492494242 + -1.59375 = -3.086244242 v = -3.086244242 Simplifying v = -3.086244242Solution
The solution to the problem is based on the solutions from the subproblems. v = {-0.101255758, -3.086244242}
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