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Simplifying -4r2 + 21r = r + 13 Reorder the terms: 21r + -4r2 = r + 13 Reorder the terms: 21r + -4r2 = 13 + r Solving 21r + -4r2 = 13 + r Solving for variable 'r'. Reorder the terms: -13 + 21r + -1r + -4r2 = 13 + r + -13 + -1r Combine like terms: 21r + -1r = 20r -13 + 20r + -4r2 = 13 + r + -13 + -1r Reorder the terms: -13 + 20r + -4r2 = 13 + -13 + r + -1r Combine like terms: 13 + -13 = 0 -13 + 20r + -4r2 = 0 + r + -1r -13 + 20r + -4r2 = r + -1r Combine like terms: r + -1r = 0 -13 + 20r + -4r2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 3.25 + -5r + r2 = 0 Move the constant term to the right: Add '-3.25' to each side of the equation. 3.25 + -5r + -3.25 + r2 = 0 + -3.25 Reorder the terms: 3.25 + -3.25 + -5r + r2 = 0 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + -5r + r2 = 0 + -3.25 -5r + r2 = 0 + -3.25 Combine like terms: 0 + -3.25 = -3.25 -5r + r2 = -3.25 The r term is -5r. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5r + 6.25 + r2 = -3.25 + 6.25 Reorder the terms: 6.25 + -5r + r2 = -3.25 + 6.25 Combine like terms: -3.25 + 6.25 = 3 6.25 + -5r + r2 = 3 Factor a perfect square on the left side: (r + -2.5)(r + -2.5) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (r + -2.5) equal to 1.732050808 and -1.732050808.Subproblem 1
r + -2.5 = 1.732050808 Simplifying r + -2.5 = 1.732050808 Reorder the terms: -2.5 + r = 1.732050808 Solving -2.5 + r = 1.732050808 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + r = 1.732050808 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + r = 1.732050808 + 2.5 r = 1.732050808 + 2.5 Combine like terms: 1.732050808 + 2.5 = 4.232050808 r = 4.232050808 Simplifying r = 4.232050808Subproblem 2
r + -2.5 = -1.732050808 Simplifying r + -2.5 = -1.732050808 Reorder the terms: -2.5 + r = -1.732050808 Solving -2.5 + r = -1.732050808 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + r = -1.732050808 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + r = -1.732050808 + 2.5 r = -1.732050808 + 2.5 Combine like terms: -1.732050808 + 2.5 = 0.767949192 r = 0.767949192 Simplifying r = 0.767949192Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.232050808, 0.767949192}
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