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Simplifying 4r2 + -9r + -19 = 0 Reorder the terms: -19 + -9r + 4r2 = 0 Solving -19 + -9r + 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'. -4.75 + -2.25r + r2 = 0 Move the constant term to the right: Add '4.75' to each side of the equation. -4.75 + -2.25r + 4.75 + r2 = 0 + 4.75 Reorder the terms: -4.75 + 4.75 + -2.25r + r2 = 0 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + -2.25r + r2 = 0 + 4.75 -2.25r + r2 = 0 + 4.75 Combine like terms: 0 + 4.75 = 4.75 -2.25r + r2 = 4.75 The r term is -2.25r. Take half its coefficient (-1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. -2.25r + 1.265625 + r2 = 4.75 + 1.265625 Reorder the terms: 1.265625 + -2.25r + r2 = 4.75 + 1.265625 Combine like terms: 4.75 + 1.265625 = 6.015625 1.265625 + -2.25r + r2 = 6.015625 Factor a perfect square on the left side: (r + -1.125)(r + -1.125) = 6.015625 Calculate the square root of the right side: 2.452677109 Break this problem into two subproblems by setting (r + -1.125) equal to 2.452677109 and -2.452677109.Subproblem 1
r + -1.125 = 2.452677109 Simplifying r + -1.125 = 2.452677109 Reorder the terms: -1.125 + r = 2.452677109 Solving -1.125 + r = 2.452677109 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + r = 2.452677109 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + r = 2.452677109 + 1.125 r = 2.452677109 + 1.125 Combine like terms: 2.452677109 + 1.125 = 3.577677109 r = 3.577677109 Simplifying r = 3.577677109Subproblem 2
r + -1.125 = -2.452677109 Simplifying r + -1.125 = -2.452677109 Reorder the terms: -1.125 + r = -2.452677109 Solving -1.125 + r = -2.452677109 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + r = -2.452677109 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + r = -2.452677109 + 1.125 r = -2.452677109 + 1.125 Combine like terms: -2.452677109 + 1.125 = -1.327677109 r = -1.327677109 Simplifying r = -1.327677109Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.577677109, -1.327677109}
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