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Simplifying 24r3 + -113r2 + 64r = 0 Reorder the terms: 64r + -113r2 + 24r3 = 0 Solving 64r + -113r2 + 24r3 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), 'r'. r(64 + -113r + 24r2) = 0Subproblem 1
Set the factor 'r' equal to zero and attempt to solve: Simplifying r = 0 Solving r = 0 Move all terms containing r to the left, all other terms to the right. Simplifying r = 0Subproblem 2
Set the factor '(64 + -113r + 24r2)' equal to zero and attempt to solve: Simplifying 64 + -113r + 24r2 = 0 Solving 64 + -113r + 24r2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. 2.666666667 + -4.708333333r + r2 = 0 Move the constant term to the right: Add '-2.666666667' to each side of the equation. 2.666666667 + -4.708333333r + -2.666666667 + r2 = 0 + -2.666666667 Reorder the terms: 2.666666667 + -2.666666667 + -4.708333333r + r2 = 0 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + -4.708333333r + r2 = 0 + -2.666666667 -4.708333333r + r2 = 0 + -2.666666667 Combine like terms: 0 + -2.666666667 = -2.666666667 -4.708333333r + r2 = -2.666666667 The r term is -4.708333333r. Take half its coefficient (-2.354166667). Square it (5.542100696) and add it to both sides. Add '5.542100696' to each side of the equation. -4.708333333r + 5.542100696 + r2 = -2.666666667 + 5.542100696 Reorder the terms: 5.542100696 + -4.708333333r + r2 = -2.666666667 + 5.542100696 Combine like terms: -2.666666667 + 5.542100696 = 2.875434029 5.542100696 + -4.708333333r + r2 = 2.875434029 Factor a perfect square on the left side: (r + -2.354166667)(r + -2.354166667) = 2.875434029 Calculate the square root of the right side: 1.695710479 Break this problem into two subproblems by setting (r + -2.354166667) equal to 1.695710479 and -1.695710479.Subproblem 1
r + -2.354166667 = 1.695710479 Simplifying r + -2.354166667 = 1.695710479 Reorder the terms: -2.354166667 + r = 1.695710479 Solving -2.354166667 + r = 1.695710479 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.354166667' to each side of the equation. -2.354166667 + 2.354166667 + r = 1.695710479 + 2.354166667 Combine like terms: -2.354166667 + 2.354166667 = 0.000000000 0.000000000 + r = 1.695710479 + 2.354166667 r = 1.695710479 + 2.354166667 Combine like terms: 1.695710479 + 2.354166667 = 4.049877146 r = 4.049877146 Simplifying r = 4.049877146Subproblem 2
r + -2.354166667 = -1.695710479 Simplifying r + -2.354166667 = -1.695710479 Reorder the terms: -2.354166667 + r = -1.695710479 Solving -2.354166667 + r = -1.695710479 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.354166667' to each side of the equation. -2.354166667 + 2.354166667 + r = -1.695710479 + 2.354166667 Combine like terms: -2.354166667 + 2.354166667 = 0.000000000 0.000000000 + r = -1.695710479 + 2.354166667 r = -1.695710479 + 2.354166667 Combine like terms: -1.695710479 + 2.354166667 = 0.658456188 r = 0.658456188 Simplifying r = 0.658456188Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.049877146, 0.658456188}Solution
r = {0, 4.049877146, 0.658456188}
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