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Simplifying 6r2 + -9r = 1 Reorder the terms: -9r + 6r2 = 1 Solving -9r + 6r2 = 1 Solving for variable 'r'. Reorder the terms: -1 + -9r + 6r2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -9r + 6r2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.1666666667 + -1.5r + r2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + -1.5r + 0.1666666667 + r2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + -1.5r + r2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + -1.5r + r2 = 0 + 0.1666666667 -1.5r + r2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 -1.5r + r2 = 0.1666666667 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 = 0.1666666667 + 0.5625 Reorder the terms: 0.5625 + -1.5r + r2 = 0.1666666667 + 0.5625 Combine like terms: 0.1666666667 + 0.5625 = 0.7291666667 0.5625 + -1.5r + r2 = 0.7291666667 Factor a perfect square on the left side: (r + -0.75)(r + -0.75) = 0.7291666667 Calculate the square root of the right side: 0.853912564 Break this problem into two subproblems by setting (r + -0.75) equal to 0.853912564 and -0.853912564.Subproblem 1
r + -0.75 = 0.853912564 Simplifying r + -0.75 = 0.853912564 Reorder the terms: -0.75 + r = 0.853912564 Solving -0.75 + r = 0.853912564 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 = 0.853912564 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = 0.853912564 + 0.75 r = 0.853912564 + 0.75 Combine like terms: 0.853912564 + 0.75 = 1.603912564 r = 1.603912564 Simplifying r = 1.603912564Subproblem 2
r + -0.75 = -0.853912564 Simplifying r + -0.75 = -0.853912564 Reorder the terms: -0.75 + r = -0.853912564 Solving -0.75 + r = -0.853912564 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 = -0.853912564 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = -0.853912564 + 0.75 r = -0.853912564 + 0.75 Combine like terms: -0.853912564 + 0.75 = -0.103912564 r = -0.103912564 Simplifying r = -0.103912564Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.603912564, -0.103912564}
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