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Simplifying 4r2 = 11r + 19 Reorder the terms: 4r2 = 19 + 11r Solving 4r2 = 19 + 11r Solving for variable 'r'. Reorder the terms: -19 + -11r + 4r2 = 19 + 11r + -19 + -11r Reorder the terms: -19 + -11r + 4r2 = 19 + -19 + 11r + -11r Combine like terms: 19 + -19 = 0 -19 + -11r + 4r2 = 0 + 11r + -11r -19 + -11r + 4r2 = 11r + -11r Combine like terms: 11r + -11r = 0 -19 + -11r + 4r2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -4.75 + -2.75r + r2 = 0 Move the constant term to the right: Add '4.75' to each side of the equation. -4.75 + -2.75r + 4.75 + r2 = 0 + 4.75 Reorder the terms: -4.75 + 4.75 + -2.75r + r2 = 0 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + -2.75r + r2 = 0 + 4.75 -2.75r + r2 = 0 + 4.75 Combine like terms: 0 + 4.75 = 4.75 -2.75r + r2 = 4.75 The r term is -2.75r. Take half its coefficient (-1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. -2.75r + 1.890625 + r2 = 4.75 + 1.890625 Reorder the terms: 1.890625 + -2.75r + r2 = 4.75 + 1.890625 Combine like terms: 4.75 + 1.890625 = 6.640625 1.890625 + -2.75r + r2 = 6.640625 Factor a perfect square on the left side: (r + -1.375)(r + -1.375) = 6.640625 Calculate the square root of the right side: 2.576941016 Break this problem into two subproblems by setting (r + -1.375) equal to 2.576941016 and -2.576941016.Subproblem 1
r + -1.375 = 2.576941016 Simplifying r + -1.375 = 2.576941016 Reorder the terms: -1.375 + r = 2.576941016 Solving -1.375 + r = 2.576941016 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + r = 2.576941016 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + r = 2.576941016 + 1.375 r = 2.576941016 + 1.375 Combine like terms: 2.576941016 + 1.375 = 3.951941016 r = 3.951941016 Simplifying r = 3.951941016Subproblem 2
r + -1.375 = -2.576941016 Simplifying r + -1.375 = -2.576941016 Reorder the terms: -1.375 + r = -2.576941016 Solving -1.375 + r = -2.576941016 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + r = -2.576941016 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + r = -2.576941016 + 1.375 r = -2.576941016 + 1.375 Combine like terms: -2.576941016 + 1.375 = -1.201941016 r = -1.201941016 Simplifying r = -1.201941016Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.951941016, -1.201941016}
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