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Simplifying 4r2 = 0.6r + 0.5 Reorder the terms: 4r2 = 0.5 + 0.6r Solving 4r2 = 0.5 + 0.6r Solving for variable 'r'. Reorder the terms: -0.5 + -0.6r + 4r2 = 0.5 + 0.6r + -0.5 + -0.6r Reorder the terms: -0.5 + -0.6r + 4r2 = 0.5 + -0.5 + 0.6r + -0.6r Combine like terms: 0.5 + -0.5 = 0.0 -0.5 + -0.6r + 4r2 = 0.0 + 0.6r + -0.6r -0.5 + -0.6r + 4r2 = 0.6r + -0.6r Combine like terms: 0.6r + -0.6r = 0.0 -0.5 + -0.6r + 4r2 = 0.0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.125 + -0.15r + r2 = 0.0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + -0.15r + 0.125 + r2 = 0.0 + 0.125 Reorder the terms: -0.125 + 0.125 + -0.15r + r2 = 0.0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + -0.15r + r2 = 0.0 + 0.125 -0.15r + r2 = 0.0 + 0.125 Combine like terms: 0.0 + 0.125 = 0.125 -0.15r + r2 = 0.125 The r term is -0.15r. Take half its coefficient (-0.075). Square it (0.005625) and add it to both sides. Add '0.005625' to each side of the equation. -0.15r + 0.005625 + r2 = 0.125 + 0.005625 Reorder the terms: 0.005625 + -0.15r + r2 = 0.125 + 0.005625 Combine like terms: 0.125 + 0.005625 = 0.130625 0.005625 + -0.15r + r2 = 0.130625 Factor a perfect square on the left side: (r + -0.075)(r + -0.075) = 0.130625 Calculate the square root of the right side: 0.361420807 Break this problem into two subproblems by setting (r + -0.075) equal to 0.361420807 and -0.361420807.Subproblem 1
r + -0.075 = 0.361420807 Simplifying r + -0.075 = 0.361420807 Reorder the terms: -0.075 + r = 0.361420807 Solving -0.075 + r = 0.361420807 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.075' to each side of the equation. -0.075 + 0.075 + r = 0.361420807 + 0.075 Combine like terms: -0.075 + 0.075 = 0.000 0.000 + r = 0.361420807 + 0.075 r = 0.361420807 + 0.075 Combine like terms: 0.361420807 + 0.075 = 0.436420807 r = 0.436420807 Simplifying r = 0.436420807Subproblem 2
r + -0.075 = -0.361420807 Simplifying r + -0.075 = -0.361420807 Reorder the terms: -0.075 + r = -0.361420807 Solving -0.075 + r = -0.361420807 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.075' to each side of the equation. -0.075 + 0.075 + r = -0.361420807 + 0.075 Combine like terms: -0.075 + 0.075 = 0.000 0.000 + r = -0.361420807 + 0.075 r = -0.361420807 + 0.075 Combine like terms: -0.361420807 + 0.075 = -0.286420807 r = -0.286420807 Simplifying r = -0.286420807Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.436420807, -0.286420807}
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