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Simplifying 2r2 + 6r + -6 = 0 Reorder the terms: -6 + 6r + 2r2 = 0 Solving -6 + 6r + 2r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + 3r + r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + 3r + r2)' equal to zero and attempt to solve: Simplifying -3 + 3r + r2 = 0 Solving -3 + 3r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + 3r + 3 + r2 = 0 + 3 Reorder the terms: -3 + 3 + 3r + r2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 3r + r2 = 0 + 3 3r + r2 = 0 + 3 Combine like terms: 0 + 3 = 3 3r + r2 = 3 The r term is 3r. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3r + 2.25 + r2 = 3 + 2.25 Reorder the terms: 2.25 + 3r + r2 = 3 + 2.25 Combine like terms: 3 + 2.25 = 5.25 2.25 + 3r + r2 = 5.25 Factor a perfect square on the left side: (r + 1.5)(r + 1.5) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (r + 1.5) equal to 2.291287847 and -2.291287847.Subproblem 1
r + 1.5 = 2.291287847 Simplifying r + 1.5 = 2.291287847 Reorder the terms: 1.5 + r = 2.291287847 Solving 1.5 + r = 2.291287847 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = 2.291287847 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = 2.291287847 + -1.5 r = 2.291287847 + -1.5 Combine like terms: 2.291287847 + -1.5 = 0.791287847 r = 0.791287847 Simplifying r = 0.791287847Subproblem 2
r + 1.5 = -2.291287847 Simplifying r + 1.5 = -2.291287847 Reorder the terms: 1.5 + r = -2.291287847 Solving 1.5 + r = -2.291287847 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = -2.291287847 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = -2.291287847 + -1.5 r = -2.291287847 + -1.5 Combine like terms: -2.291287847 + -1.5 = -3.791287847 r = -3.791287847 Simplifying r = -3.791287847Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.791287847, -3.791287847}Solution
r = {0.791287847, -3.791287847}
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