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Simplifying (4r + -12)(7r + -16) = -1 Reorder the terms: (-12 + 4r)(7r + -16) = -1 Reorder the terms: (-12 + 4r)(-16 + 7r) = -1 Multiply (-12 + 4r) * (-16 + 7r) (-12(-16 + 7r) + 4r * (-16 + 7r)) = -1 ((-16 * -12 + 7r * -12) + 4r * (-16 + 7r)) = -1 ((192 + -84r) + 4r * (-16 + 7r)) = -1 (192 + -84r + (-16 * 4r + 7r * 4r)) = -1 (192 + -84r + (-64r + 28r2)) = -1 Combine like terms: -84r + -64r = -148r (192 + -148r + 28r2) = -1 Solving 192 + -148r + 28r2 = -1 Solving for variable 'r'. Reorder the terms: 192 + 1 + -148r + 28r2 = -1 + 1 Combine like terms: 192 + 1 = 193 193 + -148r + 28r2 = -1 + 1 Combine like terms: -1 + 1 = 0 193 + -148r + 28r2 = 0 Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. 6.892857143 + -5.285714286r + r2 = 0 Move the constant term to the right: Add '-6.892857143' to each side of the equation. 6.892857143 + -5.285714286r + -6.892857143 + r2 = 0 + -6.892857143 Reorder the terms: 6.892857143 + -6.892857143 + -5.285714286r + r2 = 0 + -6.892857143 Combine like terms: 6.892857143 + -6.892857143 = 0.000000000 0.000000000 + -5.285714286r + r2 = 0 + -6.892857143 -5.285714286r + r2 = 0 + -6.892857143 Combine like terms: 0 + -6.892857143 = -6.892857143 -5.285714286r + r2 = -6.892857143 The r term is -5.285714286r. Take half its coefficient (-2.642857143). Square it (6.984693878) and add it to both sides. Add '6.984693878' to each side of the equation. -5.285714286r + 6.984693878 + r2 = -6.892857143 + 6.984693878 Reorder the terms: 6.984693878 + -5.285714286r + r2 = -6.892857143 + 6.984693878 Combine like terms: -6.892857143 + 6.984693878 = 0.091836735 6.984693878 + -5.285714286r + r2 = 0.091836735 Factor a perfect square on the left side: (r + -2.642857143)(r + -2.642857143) = 0.091836735 Calculate the square root of the right side: 0.303045764 Break this problem into two subproblems by setting (r + -2.642857143) equal to 0.303045764 and -0.303045764.Subproblem 1
r + -2.642857143 = 0.303045764 Simplifying r + -2.642857143 = 0.303045764 Reorder the terms: -2.642857143 + r = 0.303045764 Solving -2.642857143 + r = 0.303045764 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.642857143' to each side of the equation. -2.642857143 + 2.642857143 + r = 0.303045764 + 2.642857143 Combine like terms: -2.642857143 + 2.642857143 = 0.000000000 0.000000000 + r = 0.303045764 + 2.642857143 r = 0.303045764 + 2.642857143 Combine like terms: 0.303045764 + 2.642857143 = 2.945902907 r = 2.945902907 Simplifying r = 2.945902907Subproblem 2
r + -2.642857143 = -0.303045764 Simplifying r + -2.642857143 = -0.303045764 Reorder the terms: -2.642857143 + r = -0.303045764 Solving -2.642857143 + r = -0.303045764 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.642857143' to each side of the equation. -2.642857143 + 2.642857143 + r = -0.303045764 + 2.642857143 Combine like terms: -2.642857143 + 2.642857143 = 0.000000000 0.000000000 + r = -0.303045764 + 2.642857143 r = -0.303045764 + 2.642857143 Combine like terms: -0.303045764 + 2.642857143 = 2.339811379 r = 2.339811379 Simplifying r = 2.339811379Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.945902907, 2.339811379}
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