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Simplifying (R + -2)(R + 12) = 1 Reorder the terms: (-2 + R)(R + 12) = 1 Reorder the terms: (-2 + R)(12 + R) = 1 Multiply (-2 + R) * (12 + R) (-2(12 + R) + R(12 + R)) = 1 ((12 * -2 + R * -2) + R(12 + R)) = 1 ((-24 + -2R) + R(12 + R)) = 1 (-24 + -2R + (12 * R + R * R)) = 1 (-24 + -2R + (12R + R2)) = 1 Combine like terms: -2R + 12R = 10R (-24 + 10R + R2) = 1 Solving -24 + 10R + R2 = 1 Solving for variable 'R'. Reorder the terms: -24 + -1 + 10R + R2 = 1 + -1 Combine like terms: -24 + -1 = -25 -25 + 10R + R2 = 1 + -1 Combine like terms: 1 + -1 = 0 -25 + 10R + R2 = 0 Begin completing the square. Move the constant term to the right: Add '25' to each side of the equation. -25 + 10R + 25 + R2 = 0 + 25 Reorder the terms: -25 + 25 + 10R + R2 = 0 + 25 Combine like terms: -25 + 25 = 0 0 + 10R + R2 = 0 + 25 10R + R2 = 0 + 25 Combine like terms: 0 + 25 = 25 10R + R2 = 25 The R term is 10R. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10R + 25 + R2 = 25 + 25 Reorder the terms: 25 + 10R + R2 = 25 + 25 Combine like terms: 25 + 25 = 50 25 + 10R + R2 = 50 Factor a perfect square on the left side: (R + 5)(R + 5) = 50 Calculate the square root of the right side: 7.071067812 Break this problem into two subproblems by setting (R + 5) equal to 7.071067812 and -7.071067812.Subproblem 1
R + 5 = 7.071067812 Simplifying R + 5 = 7.071067812 Reorder the terms: 5 + R = 7.071067812 Solving 5 + R = 7.071067812 Solving for variable 'R'. Move all terms containing R to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + R = 7.071067812 + -5 Combine like terms: 5 + -5 = 0 0 + R = 7.071067812 + -5 R = 7.071067812 + -5 Combine like terms: 7.071067812 + -5 = 2.071067812 R = 2.071067812 Simplifying R = 2.071067812Subproblem 2
R + 5 = -7.071067812 Simplifying R + 5 = -7.071067812 Reorder the terms: 5 + R = -7.071067812 Solving 5 + R = -7.071067812 Solving for variable 'R'. Move all terms containing R to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + R = -7.071067812 + -5 Combine like terms: 5 + -5 = 0 0 + R = -7.071067812 + -5 R = -7.071067812 + -5 Combine like terms: -7.071067812 + -5 = -12.071067812 R = -12.071067812 Simplifying R = -12.071067812Solution
The solution to the problem is based on the solutions from the subproblems. R = {2.071067812, -12.071067812}
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