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Simplifying 2r2 + 15r + -19 = 0 Reorder the terms: -19 + 15r + 2r2 = 0 Solving -19 + 15r + 2r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9.5 + 7.5r + r2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + 7.5r + 9.5 + r2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + 7.5r + r2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + 7.5r + r2 = 0 + 9.5 7.5r + r2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 7.5r + r2 = 9.5 The r term is 7.5r. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5r + 14.0625 + r2 = 9.5 + 14.0625 Reorder the terms: 14.0625 + 7.5r + r2 = 9.5 + 14.0625 Combine like terms: 9.5 + 14.0625 = 23.5625 14.0625 + 7.5r + r2 = 23.5625 Factor a perfect square on the left side: (r + 3.75)(r + 3.75) = 23.5625 Calculate the square root of the right side: 4.85412196 Break this problem into two subproblems by setting (r + 3.75) equal to 4.85412196 and -4.85412196.Subproblem 1
r + 3.75 = 4.85412196 Simplifying r + 3.75 = 4.85412196 Reorder the terms: 3.75 + r = 4.85412196 Solving 3.75 + r = 4.85412196 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + r = 4.85412196 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + r = 4.85412196 + -3.75 r = 4.85412196 + -3.75 Combine like terms: 4.85412196 + -3.75 = 1.10412196 r = 1.10412196 Simplifying r = 1.10412196Subproblem 2
r + 3.75 = -4.85412196 Simplifying r + 3.75 = -4.85412196 Reorder the terms: 3.75 + r = -4.85412196 Solving 3.75 + r = -4.85412196 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + r = -4.85412196 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + r = -4.85412196 + -3.75 r = -4.85412196 + -3.75 Combine like terms: -4.85412196 + -3.75 = -8.60412196 r = -8.60412196 Simplifying r = -8.60412196Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.10412196, -8.60412196}
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