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Simplifying 5r2 + -13r + 19 = 3r2 Reorder the terms: 19 + -13r + 5r2 = 3r2 Solving 19 + -13r + 5r2 = 3r2 Solving for variable 'r'. Combine like terms: 5r2 + -3r2 = 2r2 19 + -13r + 2r2 = 3r2 + -3r2 Combine like terms: 3r2 + -3r2 = 0 19 + -13r + 2r2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 9.5 + -6.5r + r2 = 0 Move the constant term to the right: Add '-9.5' to each side of the equation. 9.5 + -6.5r + -9.5 + r2 = 0 + -9.5 Reorder the terms: 9.5 + -9.5 + -6.5r + r2 = 0 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + -6.5r + r2 = 0 + -9.5 -6.5r + r2 = 0 + -9.5 Combine like terms: 0 + -9.5 = -9.5 -6.5r + r2 = -9.5 The r term is -6.5r. Take half its coefficient (-3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. -6.5r + 10.5625 + r2 = -9.5 + 10.5625 Reorder the terms: 10.5625 + -6.5r + r2 = -9.5 + 10.5625 Combine like terms: -9.5 + 10.5625 = 1.0625 10.5625 + -6.5r + r2 = 1.0625 Factor a perfect square on the left side: (r + -3.25)(r + -3.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (r + -3.25) equal to 1.030776406 and -1.030776406.Subproblem 1
r + -3.25 = 1.030776406 Simplifying r + -3.25 = 1.030776406 Reorder the terms: -3.25 + r = 1.030776406 Solving -3.25 + r = 1.030776406 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + r = 1.030776406 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + r = 1.030776406 + 3.25 r = 1.030776406 + 3.25 Combine like terms: 1.030776406 + 3.25 = 4.280776406 r = 4.280776406 Simplifying r = 4.280776406Subproblem 2
r + -3.25 = -1.030776406 Simplifying r + -3.25 = -1.030776406 Reorder the terms: -3.25 + r = -1.030776406 Solving -3.25 + r = -1.030776406 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + r = -1.030776406 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + r = -1.030776406 + 3.25 r = -1.030776406 + 3.25 Combine like terms: -1.030776406 + 3.25 = 2.219223594 r = 2.219223594 Simplifying r = 2.219223594Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.280776406, 2.219223594}
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