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Simplifying 4r2 + -23r + -40 = 0 Reorder the terms: -40 + -23r + 4r2 = 0 Solving -40 + -23r + 4r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -10 + -5.75r + r2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + -5.75r + 10 + r2 = 0 + 10 Reorder the terms: -10 + 10 + -5.75r + r2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + -5.75r + r2 = 0 + 10 -5.75r + r2 = 0 + 10 Combine like terms: 0 + 10 = 10 -5.75r + r2 = 10 The r term is -5.75r. Take half its coefficient (-2.875). Square it (8.265625) and add it to both sides. Add '8.265625' to each side of the equation. -5.75r + 8.265625 + r2 = 10 + 8.265625 Reorder the terms: 8.265625 + -5.75r + r2 = 10 + 8.265625 Combine like terms: 10 + 8.265625 = 18.265625 8.265625 + -5.75r + r2 = 18.265625 Factor a perfect square on the left side: (r + -2.875)(r + -2.875) = 18.265625 Calculate the square root of the right side: 4.273830249 Break this problem into two subproblems by setting (r + -2.875) equal to 4.273830249 and -4.273830249.Subproblem 1
r + -2.875 = 4.273830249 Simplifying r + -2.875 = 4.273830249 Reorder the terms: -2.875 + r = 4.273830249 Solving -2.875 + r = 4.273830249 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.875' to each side of the equation. -2.875 + 2.875 + r = 4.273830249 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + r = 4.273830249 + 2.875 r = 4.273830249 + 2.875 Combine like terms: 4.273830249 + 2.875 = 7.148830249 r = 7.148830249 Simplifying r = 7.148830249Subproblem 2
r + -2.875 = -4.273830249 Simplifying r + -2.875 = -4.273830249 Reorder the terms: -2.875 + r = -4.273830249 Solving -2.875 + r = -4.273830249 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.875' to each side of the equation. -2.875 + 2.875 + r = -4.273830249 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + r = -4.273830249 + 2.875 r = -4.273830249 + 2.875 Combine like terms: -4.273830249 + 2.875 = -1.398830249 r = -1.398830249 Simplifying r = -1.398830249Solution
The solution to the problem is based on the solutions from the subproblems. r = {7.148830249, -1.398830249}
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