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Simplifying -6r2 + 9r + 4 = 0 Reorder the terms: 4 + 9r + -6r2 = 0 Solving 4 + 9r + -6r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -0.6666666667 + -1.5r + r2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -1.5r + 0.6666666667 + r2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -1.5r + r2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -1.5r + r2 = 0 + 0.6666666667 -1.5r + r2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -1.5r + r2 = 0.6666666667 The r term is -1.5r. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5r + 0.5625 + r2 = 0.6666666667 + 0.5625 Reorder the terms: 0.5625 + -1.5r + r2 = 0.6666666667 + 0.5625 Combine like terms: 0.6666666667 + 0.5625 = 1.2291666667 0.5625 + -1.5r + r2 = 1.2291666667 Factor a perfect square on the left side: (r + -0.75)(r + -0.75) = 1.2291666667 Calculate the square root of the right side: 1.108677891 Break this problem into two subproblems by setting (r + -0.75) equal to 1.108677891 and -1.108677891.Subproblem 1
r + -0.75 = 1.108677891 Simplifying r + -0.75 = 1.108677891 Reorder the terms: -0.75 + r = 1.108677891 Solving -0.75 + r = 1.108677891 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + r = 1.108677891 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = 1.108677891 + 0.75 r = 1.108677891 + 0.75 Combine like terms: 1.108677891 + 0.75 = 1.858677891 r = 1.858677891 Simplifying r = 1.858677891Subproblem 2
r + -0.75 = -1.108677891 Simplifying r + -0.75 = -1.108677891 Reorder the terms: -0.75 + r = -1.108677891 Solving -0.75 + r = -1.108677891 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + r = -1.108677891 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = -1.108677891 + 0.75 r = -1.108677891 + 0.75 Combine like terms: -1.108677891 + 0.75 = -0.358677891 r = -0.358677891 Simplifying r = -0.358677891Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.858677891, -0.358677891}
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