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Simplifying 6r2 + -12r + -20 = -9 Reorder the terms: -20 + -12r + 6r2 = -9 Solving -20 + -12r + 6r2 = -9 Solving for variable 'r'. Reorder the terms: -20 + 9 + -12r + 6r2 = -9 + 9 Combine like terms: -20 + 9 = -11 -11 + -12r + 6r2 = -9 + 9 Combine like terms: -9 + 9 = 0 -11 + -12r + 6r2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.833333333 + -2r + r2 = 0 Move the constant term to the right: Add '1.833333333' to each side of the equation. -1.833333333 + -2r + 1.833333333 + r2 = 0 + 1.833333333 Reorder the terms: -1.833333333 + 1.833333333 + -2r + r2 = 0 + 1.833333333 Combine like terms: -1.833333333 + 1.833333333 = 0.000000000 0.000000000 + -2r + r2 = 0 + 1.833333333 -2r + r2 = 0 + 1.833333333 Combine like terms: 0 + 1.833333333 = 1.833333333 -2r + r2 = 1.833333333 The r term is -2r. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2r + 1 + r2 = 1.833333333 + 1 Reorder the terms: 1 + -2r + r2 = 1.833333333 + 1 Combine like terms: 1.833333333 + 1 = 2.833333333 1 + -2r + r2 = 2.833333333 Factor a perfect square on the left side: (r + -1)(r + -1) = 2.833333333 Calculate the square root of the right side: 1.683250823 Break this problem into two subproblems by setting (r + -1) equal to 1.683250823 and -1.683250823.Subproblem 1
r + -1 = 1.683250823 Simplifying r + -1 = 1.683250823 Reorder the terms: -1 + r = 1.683250823 Solving -1 + r = 1.683250823 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = 1.683250823 + 1 Combine like terms: -1 + 1 = 0 0 + r = 1.683250823 + 1 r = 1.683250823 + 1 Combine like terms: 1.683250823 + 1 = 2.683250823 r = 2.683250823 Simplifying r = 2.683250823Subproblem 2
r + -1 = -1.683250823 Simplifying r + -1 = -1.683250823 Reorder the terms: -1 + r = -1.683250823 Solving -1 + r = -1.683250823 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = -1.683250823 + 1 Combine like terms: -1 + 1 = 0 0 + r = -1.683250823 + 1 r = -1.683250823 + 1 Combine like terms: -1.683250823 + 1 = -0.683250823 r = -0.683250823 Simplifying r = -0.683250823Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.683250823, -0.683250823}
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