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Simplifying 6r2 + 2r + -24 = 0 Reorder the terms: -24 + 2r + 6r2 = 0 Solving -24 + 2r + 6r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2'. 2(-12 + r + 3r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-12 + r + 3r2)' equal to zero and attempt to solve: Simplifying -12 + r + 3r2 = 0 Solving -12 + r + 3r2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -4 + 0.3333333333r + r2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 0.3333333333r + 4 + r2 = 0 + 4 Reorder the terms: -4 + 4 + 0.3333333333r + r2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 0.3333333333r + r2 = 0 + 4 0.3333333333r + r2 = 0 + 4 Combine like terms: 0 + 4 = 4 0.3333333333r + r2 = 4 The r term is r. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.3333333333r + 0.25 + r2 = 4 + 0.25 Reorder the terms: 0.25 + 0.3333333333r + r2 = 4 + 0.25 Combine like terms: 4 + 0.25 = 4.25 0.25 + 0.3333333333r + r2 = 4.25 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (r + 0.5) equal to 2.061552813 and -2.061552813.Subproblem 1
r + 0.5 = 2.061552813 Simplifying r + 0.5 = 2.061552813 Reorder the terms: 0.5 + r = 2.061552813 Solving 0.5 + r = 2.061552813 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 2.061552813 + -0.5 r = 2.061552813 + -0.5 Combine like terms: 2.061552813 + -0.5 = 1.561552813 r = 1.561552813 Simplifying r = 1.561552813Subproblem 2
r + 0.5 = -2.061552813 Simplifying r + 0.5 = -2.061552813 Reorder the terms: 0.5 + r = -2.061552813 Solving 0.5 + r = -2.061552813 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -2.061552813 + -0.5 r = -2.061552813 + -0.5 Combine like terms: -2.061552813 + -0.5 = -2.561552813 r = -2.561552813 Simplifying r = -2.561552813Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.561552813, -2.561552813}Solution
r = {1.561552813, -2.561552813}
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