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Simplifying 15r2 + -7r + -12 = 0 Reorder the terms: -12 + -7r + 15r2 = 0 Solving -12 + -7r + 15r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.8 + -0.4666666667r + r2 = 0 Move the constant term to the right: Add '0.8' to each side of the equation. -0.8 + -0.4666666667r + 0.8 + r2 = 0 + 0.8 Reorder the terms: -0.8 + 0.8 + -0.4666666667r + r2 = 0 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + -0.4666666667r + r2 = 0 + 0.8 -0.4666666667r + r2 = 0 + 0.8 Combine like terms: 0 + 0.8 = 0.8 -0.4666666667r + r2 = 0.8 The r term is -0.4666666667r. Take half its coefficient (-0.2333333334). Square it (0.05444444448) and add it to both sides. Add '0.05444444448' to each side of the equation. -0.4666666667r + 0.05444444448 + r2 = 0.8 + 0.05444444448 Reorder the terms: 0.05444444448 + -0.4666666667r + r2 = 0.8 + 0.05444444448 Combine like terms: 0.8 + 0.05444444448 = 0.85444444448 0.05444444448 + -0.4666666667r + r2 = 0.85444444448 Factor a perfect square on the left side: (r + -0.2333333334)(r + -0.2333333334) = 0.85444444448 Calculate the square root of the right side: 0.924361642 Break this problem into two subproblems by setting (r + -0.2333333334) equal to 0.924361642 and -0.924361642.Subproblem 1
r + -0.2333333334 = 0.924361642 Simplifying r + -0.2333333334 = 0.924361642 Reorder the terms: -0.2333333334 + r = 0.924361642 Solving -0.2333333334 + r = 0.924361642 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.2333333334' to each side of the equation. -0.2333333334 + 0.2333333334 + r = 0.924361642 + 0.2333333334 Combine like terms: -0.2333333334 + 0.2333333334 = 0.0000000000 0.0000000000 + r = 0.924361642 + 0.2333333334 r = 0.924361642 + 0.2333333334 Combine like terms: 0.924361642 + 0.2333333334 = 1.1576949754 r = 1.1576949754 Simplifying r = 1.1576949754Subproblem 2
r + -0.2333333334 = -0.924361642 Simplifying r + -0.2333333334 = -0.924361642 Reorder the terms: -0.2333333334 + r = -0.924361642 Solving -0.2333333334 + r = -0.924361642 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.2333333334' to each side of the equation. -0.2333333334 + 0.2333333334 + r = -0.924361642 + 0.2333333334 Combine like terms: -0.2333333334 + 0.2333333334 = 0.0000000000 0.0000000000 + r = -0.924361642 + 0.2333333334 r = -0.924361642 + 0.2333333334 Combine like terms: -0.924361642 + 0.2333333334 = -0.6910283086 r = -0.6910283086 Simplifying r = -0.6910283086Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.1576949754, -0.6910283086}
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