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Simplifying 2r2 + 9r + 5 = 0 Reorder the terms: 5 + 9r + 2r2 = 0 Solving 5 + 9r + 2r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 2.5 + 4.5r + r2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + 4.5r + -2.5 + r2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + 4.5r + r2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + 4.5r + r2 = 0 + -2.5 4.5r + r2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 4.5r + r2 = -2.5 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 = -2.5 + 5.0625 Reorder the terms: 5.0625 + 4.5r + r2 = -2.5 + 5.0625 Combine like terms: -2.5 + 5.0625 = 2.5625 5.0625 + 4.5r + r2 = 2.5625 Factor a perfect square on the left side: (r + 2.25)(r + 2.25) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (r + 2.25) equal to 1.600781059 and -1.600781059.Subproblem 1
r + 2.25 = 1.600781059 Simplifying r + 2.25 = 1.600781059 Reorder the terms: 2.25 + r = 1.600781059 Solving 2.25 + r = 1.600781059 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 = 1.600781059 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + r = 1.600781059 + -2.25 r = 1.600781059 + -2.25 Combine like terms: 1.600781059 + -2.25 = -0.649218941 r = -0.649218941 Simplifying r = -0.649218941Subproblem 2
r + 2.25 = -1.600781059 Simplifying r + 2.25 = -1.600781059 Reorder the terms: 2.25 + r = -1.600781059 Solving 2.25 + r = -1.600781059 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 = -1.600781059 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + r = -1.600781059 + -2.25 r = -1.600781059 + -2.25 Combine like terms: -1.600781059 + -2.25 = -3.850781059 r = -3.850781059 Simplifying r = -3.850781059Solution
The solution to the problem is based on the solutions from the subproblems. r = {-0.649218941, -3.850781059}
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