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Simplifying 2r3 + 14r2 + 16r = 0 Reorder the terms: 16r + 14r2 + 2r3 = 0 Solving 16r + 14r2 + 2r3 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2r'. 2r(8 + 7r + r2) = 0 Ignore the factor 2.Subproblem 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 '(8 + 7r + r2)' equal to zero and attempt to solve: Simplifying 8 + 7r + r2 = 0 Solving 8 + 7r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '-8' to each side of the equation. 8 + 7r + -8 + r2 = 0 + -8 Reorder the terms: 8 + -8 + 7r + r2 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + 7r + r2 = 0 + -8 7r + r2 = 0 + -8 Combine like terms: 0 + -8 = -8 7r + r2 = -8 The r term is 7r. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7r + 12.25 + r2 = -8 + 12.25 Reorder the terms: 12.25 + 7r + r2 = -8 + 12.25 Combine like terms: -8 + 12.25 = 4.25 12.25 + 7r + r2 = 4.25 Factor a perfect square on the left side: (r + 3.5)(r + 3.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (r + 3.5) equal to 2.061552813 and -2.061552813.Subproblem 1
r + 3.5 = 2.061552813 Simplifying r + 3.5 = 2.061552813 Reorder the terms: 3.5 + r = 2.061552813 Solving 3.5 + r = 2.061552813 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + r = 2.061552813 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + r = 2.061552813 + -3.5 r = 2.061552813 + -3.5 Combine like terms: 2.061552813 + -3.5 = -1.438447187 r = -1.438447187 Simplifying r = -1.438447187Subproblem 2
r + 3.5 = -2.061552813 Simplifying r + 3.5 = -2.061552813 Reorder the terms: 3.5 + r = -2.061552813 Solving 3.5 + r = -2.061552813 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + r = -2.061552813 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + r = -2.061552813 + -3.5 r = -2.061552813 + -3.5 Combine like terms: -2.061552813 + -3.5 = -5.561552813 r = -5.561552813 Simplifying r = -5.561552813Solution
The solution to the problem is based on the solutions from the subproblems. r = {-1.438447187, -5.561552813}Solution
r = {0, -1.438447187, -5.561552813}
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