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Simplifying -3.6 = r(-6r + 0.6) Reorder the terms: -3.6 = r(0.6 + -6r) -3.6 = (0.6 * r + -6r * r) -3.6 = (0.6r + -6r2) Solving -3.6 = 0.6r + -6r2 Solving for variable 'r'. Reorder the terms: -3.6 + -0.6r + 6r2 = 0.6r + -0.6r + -6r2 + 6r2 Combine like terms: 0.6r + -0.6r = 0.0 -3.6 + -0.6r + 6r2 = 0.0 + -6r2 + 6r2 -3.6 + -0.6r + 6r2 = -6r2 + 6r2 Combine like terms: -6r2 + 6r2 = 0 -3.6 + -0.6r + 6r2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.6 + -0.1r + r2 = 0 Move the constant term to the right: Add '0.6' to each side of the equation. -0.6 + -0.1r + 0.6 + r2 = 0 + 0.6 Reorder the terms: -0.6 + 0.6 + -0.1r + r2 = 0 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + -0.1r + r2 = 0 + 0.6 -0.1r + r2 = 0 + 0.6 Combine like terms: 0 + 0.6 = 0.6 -0.1r + r2 = 0.6 The r term is -0.1r. Take half its coefficient (-0.05). Square it (0.0025) and add it to both sides. Add '0.0025' to each side of the equation. -0.1r + 0.0025 + r2 = 0.6 + 0.0025 Reorder the terms: 0.0025 + -0.1r + r2 = 0.6 + 0.0025 Combine like terms: 0.6 + 0.0025 = 0.6025 0.0025 + -0.1r + r2 = 0.6025 Factor a perfect square on the left side: (r + -0.05)(r + -0.05) = 0.6025 Calculate the square root of the right side: 0.776208735 Break this problem into two subproblems by setting (r + -0.05) equal to 0.776208735 and -0.776208735.Subproblem 1
r + -0.05 = 0.776208735 Simplifying r + -0.05 = 0.776208735 Reorder the terms: -0.05 + r = 0.776208735 Solving -0.05 + r = 0.776208735 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.05' to each side of the equation. -0.05 + 0.05 + r = 0.776208735 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + r = 0.776208735 + 0.05 r = 0.776208735 + 0.05 Combine like terms: 0.776208735 + 0.05 = 0.826208735 r = 0.826208735 Simplifying r = 0.826208735Subproblem 2
r + -0.05 = -0.776208735 Simplifying r + -0.05 = -0.776208735 Reorder the terms: -0.05 + r = -0.776208735 Solving -0.05 + r = -0.776208735 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.05' to each side of the equation. -0.05 + 0.05 + r = -0.776208735 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + r = -0.776208735 + 0.05 r = -0.776208735 + 0.05 Combine like terms: -0.776208735 + 0.05 = -0.726208735 r = -0.726208735 Simplifying r = -0.726208735Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.826208735, -0.726208735}
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