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Simplifying 10 = r(r + -5) Reorder the terms: 10 = r(-5 + r) 10 = (-5 * r + r * r) 10 = (-5r + r2) Solving 10 = -5r + r2 Solving for variable 'r'. Reorder the terms: 10 + 5r + -1r2 = -5r + 5r + r2 + -1r2 Combine like terms: -5r + 5r = 0 10 + 5r + -1r2 = 0 + r2 + -1r2 10 + 5r + -1r2 = r2 + -1r2 Combine like terms: r2 + -1r2 = 0 10 + 5r + -1r2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -10 + -5r + r2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + -5r + 10 + r2 = 0 + 10 Reorder the terms: -10 + 10 + -5r + r2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + -5r + r2 = 0 + 10 -5r + r2 = 0 + 10 Combine like terms: 0 + 10 = 10 -5r + r2 = 10 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 = 10 + 6.25 Reorder the terms: 6.25 + -5r + r2 = 10 + 6.25 Combine like terms: 10 + 6.25 = 16.25 6.25 + -5r + r2 = 16.25 Factor a perfect square on the left side: (r + -2.5)(r + -2.5) = 16.25 Calculate the square root of the right side: 4.031128874 Break this problem into two subproblems by setting (r + -2.5) equal to 4.031128874 and -4.031128874.Subproblem 1
r + -2.5 = 4.031128874 Simplifying r + -2.5 = 4.031128874 Reorder the terms: -2.5 + r = 4.031128874 Solving -2.5 + r = 4.031128874 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 = 4.031128874 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + r = 4.031128874 + 2.5 r = 4.031128874 + 2.5 Combine like terms: 4.031128874 + 2.5 = 6.531128874 r = 6.531128874 Simplifying r = 6.531128874Subproblem 2
r + -2.5 = -4.031128874 Simplifying r + -2.5 = -4.031128874 Reorder the terms: -2.5 + r = -4.031128874 Solving -2.5 + r = -4.031128874 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 = -4.031128874 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + r = -4.031128874 + 2.5 r = -4.031128874 + 2.5 Combine like terms: -4.031128874 + 2.5 = -1.531128874 r = -1.531128874 Simplifying r = -1.531128874Solution
The solution to the problem is based on the solutions from the subproblems. r = {6.531128874, -1.531128874}
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