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Simplifying 4r2 + 10r = -5 Reorder the terms: 10r + 4r2 = -5 Solving 10r + 4r2 = -5 Solving for variable 'r'. Reorder the terms: 5 + 10r + 4r2 = -5 + 5 Combine like terms: -5 + 5 = 0 5 + 10r + 4r2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.25 + 2.5r + r2 = 0 Move the constant term to the right: Add '-1.25' to each side of the equation. 1.25 + 2.5r + -1.25 + r2 = 0 + -1.25 Reorder the terms: 1.25 + -1.25 + 2.5r + r2 = 0 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + 2.5r + r2 = 0 + -1.25 2.5r + r2 = 0 + -1.25 Combine like terms: 0 + -1.25 = -1.25 2.5r + r2 = -1.25 The r term is 2.5r. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5r + 1.5625 + r2 = -1.25 + 1.5625 Reorder the terms: 1.5625 + 2.5r + r2 = -1.25 + 1.5625 Combine like terms: -1.25 + 1.5625 = 0.3125 1.5625 + 2.5r + r2 = 0.3125 Factor a perfect square on the left side: (r + 1.25)(r + 1.25) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (r + 1.25) equal to 0.559016994 and -0.559016994.Subproblem 1
r + 1.25 = 0.559016994 Simplifying r + 1.25 = 0.559016994 Reorder the terms: 1.25 + r = 0.559016994 Solving 1.25 + r = 0.559016994 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + r = 0.559016994 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + r = 0.559016994 + -1.25 r = 0.559016994 + -1.25 Combine like terms: 0.559016994 + -1.25 = -0.690983006 r = -0.690983006 Simplifying r = -0.690983006Subproblem 2
r + 1.25 = -0.559016994 Simplifying r + 1.25 = -0.559016994 Reorder the terms: 1.25 + r = -0.559016994 Solving 1.25 + r = -0.559016994 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + r = -0.559016994 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + r = -0.559016994 + -1.25 r = -0.559016994 + -1.25 Combine like terms: -0.559016994 + -1.25 = -1.809016994 r = -1.809016994 Simplifying r = -1.809016994Solution
The solution to the problem is based on the solutions from the subproblems. r = {-0.690983006, -1.809016994}
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