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Simplifying 15r2 + -34r + -15 = 0 Reorder the terms: -15 + -34r + 15r2 = 0 Solving -15 + -34r + 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'. -1 + -2.266666667r + r2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -2.266666667r + 1 + r2 = 0 + 1 Reorder the terms: -1 + 1 + -2.266666667r + r2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -2.266666667r + r2 = 0 + 1 -2.266666667r + r2 = 0 + 1 Combine like terms: 0 + 1 = 1 -2.266666667r + r2 = 1 The r term is -2.266666667r. Take half its coefficient (-1.133333334). Square it (1.284444446) and add it to both sides. Add '1.284444446' to each side of the equation. -2.266666667r + 1.284444446 + r2 = 1 + 1.284444446 Reorder the terms: 1.284444446 + -2.266666667r + r2 = 1 + 1.284444446 Combine like terms: 1 + 1.284444446 = 2.284444446 1.284444446 + -2.266666667r + r2 = 2.284444446 Factor a perfect square on the left side: (r + -1.133333334)(r + -1.133333334) = 2.284444446 Calculate the square root of the right side: 1.511437874 Break this problem into two subproblems by setting (r + -1.133333334) equal to 1.511437874 and -1.511437874.Subproblem 1
r + -1.133333334 = 1.511437874 Simplifying r + -1.133333334 = 1.511437874 Reorder the terms: -1.133333334 + r = 1.511437874 Solving -1.133333334 + r = 1.511437874 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.133333334' to each side of the equation. -1.133333334 + 1.133333334 + r = 1.511437874 + 1.133333334 Combine like terms: -1.133333334 + 1.133333334 = 0.000000000 0.000000000 + r = 1.511437874 + 1.133333334 r = 1.511437874 + 1.133333334 Combine like terms: 1.511437874 + 1.133333334 = 2.644771208 r = 2.644771208 Simplifying r = 2.644771208Subproblem 2
r + -1.133333334 = -1.511437874 Simplifying r + -1.133333334 = -1.511437874 Reorder the terms: -1.133333334 + r = -1.511437874 Solving -1.133333334 + r = -1.511437874 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.133333334' to each side of the equation. -1.133333334 + 1.133333334 + r = -1.511437874 + 1.133333334 Combine like terms: -1.133333334 + 1.133333334 = 0.000000000 0.000000000 + r = -1.511437874 + 1.133333334 r = -1.511437874 + 1.133333334 Combine like terms: -1.511437874 + 1.133333334 = -0.37810454 r = -0.37810454 Simplifying r = -0.37810454Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.644771208, -0.37810454}
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