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Simplifying 4r2 + 19r + 15 = r Reorder the terms: 15 + 19r + 4r2 = r Solving 15 + 19r + 4r2 = r Solving for variable 'r'. Reorder the terms: 15 + 19r + -1r + 4r2 = r + -1r Combine like terms: 19r + -1r = 18r 15 + 18r + 4r2 = r + -1r Combine like terms: r + -1r = 0 15 + 18r + 4r2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 3.75 + 4.5r + r2 = 0 Move the constant term to the right: Add '-3.75' to each side of the equation. 3.75 + 4.5r + -3.75 + r2 = 0 + -3.75 Reorder the terms: 3.75 + -3.75 + 4.5r + r2 = 0 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + 4.5r + r2 = 0 + -3.75 4.5r + r2 = 0 + -3.75 Combine like terms: 0 + -3.75 = -3.75 4.5r + r2 = -3.75 The r term is 4.5r. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5r + 5.0625 + r2 = -3.75 + 5.0625 Reorder the terms: 5.0625 + 4.5r + r2 = -3.75 + 5.0625 Combine like terms: -3.75 + 5.0625 = 1.3125 5.0625 + 4.5r + r2 = 1.3125 Factor a perfect square on the left side: (r + 2.25)(r + 2.25) = 1.3125 Calculate the square root of the right side: 1.145643924 Break this problem into two subproblems by setting (r + 2.25) equal to 1.145643924 and -1.145643924.Subproblem 1
r + 2.25 = 1.145643924 Simplifying r + 2.25 = 1.145643924 Reorder the terms: 2.25 + r = 1.145643924 Solving 2.25 + r = 1.145643924 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + r = 1.145643924 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + r = 1.145643924 + -2.25 r = 1.145643924 + -2.25 Combine like terms: 1.145643924 + -2.25 = -1.104356076 r = -1.104356076 Simplifying r = -1.104356076Subproblem 2
r + 2.25 = -1.145643924 Simplifying r + 2.25 = -1.145643924 Reorder the terms: 2.25 + r = -1.145643924 Solving 2.25 + r = -1.145643924 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + r = -1.145643924 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + r = -1.145643924 + -2.25 r = -1.145643924 + -2.25 Combine like terms: -1.145643924 + -2.25 = -3.395643924 r = -3.395643924 Simplifying r = -3.395643924Solution
The solution to the problem is based on the solutions from the subproblems. r = {-1.104356076, -3.395643924}
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