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Simplifying 16r2 = 6 + 24r + -9 Reorder the terms: 16r2 = 6 + -9 + 24r Combine like terms: 6 + -9 = -3 16r2 = -3 + 24r Solving 16r2 = -3 + 24r Solving for variable 'r'. Reorder the terms: 3 + -24r + 16r2 = -3 + 24r + 3 + -24r Reorder the terms: 3 + -24r + 16r2 = -3 + 3 + 24r + -24r Combine like terms: -3 + 3 = 0 3 + -24r + 16r2 = 0 + 24r + -24r 3 + -24r + 16r2 = 24r + -24r Combine like terms: 24r + -24r = 0 3 + -24r + 16r2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.1875 + -1.5r + r2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + -1.5r + -0.1875 + r2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + -1.5r + r2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + -1.5r + r2 = 0 + -0.1875 -1.5r + r2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 -1.5r + r2 = -0.1875 The r term is -1.5r. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5r + 0.5625 + r2 = -0.1875 + 0.5625 Reorder the terms: 0.5625 + -1.5r + r2 = -0.1875 + 0.5625 Combine like terms: -0.1875 + 0.5625 = 0.375 0.5625 + -1.5r + r2 = 0.375 Factor a perfect square on the left side: (r + -0.75)(r + -0.75) = 0.375 Calculate the square root of the right side: 0.612372436 Break this problem into two subproblems by setting (r + -0.75) equal to 0.612372436 and -0.612372436.Subproblem 1
r + -0.75 = 0.612372436 Simplifying r + -0.75 = 0.612372436 Reorder the terms: -0.75 + r = 0.612372436 Solving -0.75 + r = 0.612372436 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + r = 0.612372436 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = 0.612372436 + 0.75 r = 0.612372436 + 0.75 Combine like terms: 0.612372436 + 0.75 = 1.362372436 r = 1.362372436 Simplifying r = 1.362372436Subproblem 2
r + -0.75 = -0.612372436 Simplifying r + -0.75 = -0.612372436 Reorder the terms: -0.75 + r = -0.612372436 Solving -0.75 + r = -0.612372436 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + r = -0.612372436 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = -0.612372436 + 0.75 r = -0.612372436 + 0.75 Combine like terms: -0.612372436 + 0.75 = 0.137627564 r = 0.137627564 Simplifying r = 0.137627564Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.362372436, 0.137627564}
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