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Simplifying -3k(1 + -6k) = 6k + 21 (1 * -3k + -6k * -3k) = 6k + 21 (-3k + 18k2) = 6k + 21 Reorder the terms: -3k + 18k2 = 21 + 6k Solving -3k + 18k2 = 21 + 6k Solving for variable 'k'. Reorder the terms: -21 + -3k + -6k + 18k2 = 21 + 6k + -21 + -6k Combine like terms: -3k + -6k = -9k -21 + -9k + 18k2 = 21 + 6k + -21 + -6k Reorder the terms: -21 + -9k + 18k2 = 21 + -21 + 6k + -6k Combine like terms: 21 + -21 = 0 -21 + -9k + 18k2 = 0 + 6k + -6k -21 + -9k + 18k2 = 6k + -6k Combine like terms: 6k + -6k = 0 -21 + -9k + 18k2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-7 + -3k + 6k2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-7 + -3k + 6k2)' equal to zero and attempt to solve: Simplifying -7 + -3k + 6k2 = 0 Solving -7 + -3k + 6k2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.166666667 + -0.5k + k2 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + -0.5k + 1.166666667 + k2 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + -0.5k + k2 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + -0.5k + k2 = 0 + 1.166666667 -0.5k + k2 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 -0.5k + k2 = 1.166666667 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 = 1.166666667 + 0.0625 Reorder the terms: 0.0625 + -0.5k + k2 = 1.166666667 + 0.0625 Combine like terms: 1.166666667 + 0.0625 = 1.229166667 0.0625 + -0.5k + k2 = 1.229166667 Factor a perfect square on the left side: (k + -0.25)(k + -0.25) = 1.229166667 Calculate the square root of the right side: 1.108677891 Break this problem into two subproblems by setting (k + -0.25) equal to 1.108677891 and -1.108677891.Subproblem 1
k + -0.25 = 1.108677891 Simplifying k + -0.25 = 1.108677891 Reorder the terms: -0.25 + k = 1.108677891 Solving -0.25 + k = 1.108677891 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 = 1.108677891 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + k = 1.108677891 + 0.25 k = 1.108677891 + 0.25 Combine like terms: 1.108677891 + 0.25 = 1.358677891 k = 1.358677891 Simplifying k = 1.358677891Subproblem 2
k + -0.25 = -1.108677891 Simplifying k + -0.25 = -1.108677891 Reorder the terms: -0.25 + k = -1.108677891 Solving -0.25 + k = -1.108677891 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 = -1.108677891 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + k = -1.108677891 + 0.25 k = -1.108677891 + 0.25 Combine like terms: -1.108677891 + 0.25 = -0.858677891 k = -0.858677891 Simplifying k = -0.858677891Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.358677891, -0.858677891}Solution
k = {1.358677891, -0.858677891}
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