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Simplifying 12k2 + -6k + -9 = 0 Reorder the terms: -9 + -6k + 12k2 = 0 Solving -9 + -6k + 12k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '3'. 3(-3 + -2k + 4k2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-3 + -2k + 4k2)' equal to zero and attempt to solve: Simplifying -3 + -2k + 4k2 = 0 Solving -3 + -2k + 4k2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -0.5k + k2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -0.5k + 0.75 + k2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -0.5k + k2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -0.5k + k2 = 0 + 0.75 -0.5k + k2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -0.5k + k2 = 0.75 The k term is -0.5k. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5k + 0.0625 + k2 = 0.75 + 0.0625 Reorder the terms: 0.0625 + -0.5k + k2 = 0.75 + 0.0625 Combine like terms: 0.75 + 0.0625 = 0.8125 0.0625 + -0.5k + k2 = 0.8125 Factor a perfect square on the left side: (k + -0.25)(k + -0.25) = 0.8125 Calculate the square root of the right side: 0.901387819 Break this problem into two subproblems by setting (k + -0.25) equal to 0.901387819 and -0.901387819.Subproblem 1
k + -0.25 = 0.901387819 Simplifying k + -0.25 = 0.901387819 Reorder the terms: -0.25 + k = 0.901387819 Solving -0.25 + k = 0.901387819 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + k = 0.901387819 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + k = 0.901387819 + 0.25 k = 0.901387819 + 0.25 Combine like terms: 0.901387819 + 0.25 = 1.151387819 k = 1.151387819 Simplifying k = 1.151387819Subproblem 2
k + -0.25 = -0.901387819 Simplifying k + -0.25 = -0.901387819 Reorder the terms: -0.25 + k = -0.901387819 Solving -0.25 + k = -0.901387819 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + k = -0.901387819 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + k = -0.901387819 + 0.25 k = -0.901387819 + 0.25 Combine like terms: -0.901387819 + 0.25 = -0.651387819 k = -0.651387819 Simplifying k = -0.651387819Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.151387819, -0.651387819}Solution
k = {1.151387819, -0.651387819}
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