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Simplifying 3k2 + 15k2 + -2k + -10 = 0 Reorder the terms: -10 + -2k + 3k2 + 15k2 = 0 Combine like terms: 3k2 + 15k2 = 18k2 -10 + -2k + 18k2 = 0 Solving -10 + -2k + 18k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + -1k + 9k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + -1k + 9k2)' equal to zero and attempt to solve: Simplifying -5 + -1k + 9k2 = 0 Solving -5 + -1k + 9k2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.5555555556 + -0.1111111111k + k2 = 0 Move the constant term to the right: Add '0.5555555556' to each side of the equation. -0.5555555556 + -0.1111111111k + 0.5555555556 + k2 = 0 + 0.5555555556 Reorder the terms: -0.5555555556 + 0.5555555556 + -0.1111111111k + k2 = 0 + 0.5555555556 Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000 0.0000000000 + -0.1111111111k + k2 = 0 + 0.5555555556 -0.1111111111k + k2 = 0 + 0.5555555556 Combine like terms: 0 + 0.5555555556 = 0.5555555556 -0.1111111111k + k2 = 0.5555555556 The k term is -0.1111111111k. Take half its coefficient (-0.05555555555). Square it (0.003086419752) and add it to both sides. Add '0.003086419752' to each side of the equation. -0.1111111111k + 0.003086419752 + k2 = 0.5555555556 + 0.003086419752 Reorder the terms: 0.003086419752 + -0.1111111111k + k2 = 0.5555555556 + 0.003086419752 Combine like terms: 0.5555555556 + 0.003086419752 = 0.558641975352 0.003086419752 + -0.1111111111k + k2 = 0.558641975352 Factor a perfect square on the left side: (k + -0.05555555555)(k + -0.05555555555) = 0.558641975352 Calculate the square root of the right side: 0.747423558 Break this problem into two subproblems by setting (k + -0.05555555555) equal to 0.747423558 and -0.747423558.Subproblem 1
k + -0.05555555555 = 0.747423558 Simplifying k + -0.05555555555 = 0.747423558 Reorder the terms: -0.05555555555 + k = 0.747423558 Solving -0.05555555555 + k = 0.747423558 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.05555555555' to each side of the equation. -0.05555555555 + 0.05555555555 + k = 0.747423558 + 0.05555555555 Combine like terms: -0.05555555555 + 0.05555555555 = 0.00000000000 0.00000000000 + k = 0.747423558 + 0.05555555555 k = 0.747423558 + 0.05555555555 Combine like terms: 0.747423558 + 0.05555555555 = 0.80297911355 k = 0.80297911355 Simplifying k = 0.80297911355Subproblem 2
k + -0.05555555555 = -0.747423558 Simplifying k + -0.05555555555 = -0.747423558 Reorder the terms: -0.05555555555 + k = -0.747423558 Solving -0.05555555555 + k = -0.747423558 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.05555555555' to each side of the equation. -0.05555555555 + 0.05555555555 + k = -0.747423558 + 0.05555555555 Combine like terms: -0.05555555555 + 0.05555555555 = 0.00000000000 0.00000000000 + k = -0.747423558 + 0.05555555555 k = -0.747423558 + 0.05555555555 Combine like terms: -0.747423558 + 0.05555555555 = -0.69186800245 k = -0.69186800245 Simplifying k = -0.69186800245Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.80297911355, -0.69186800245}Solution
k = {0.80297911355, -0.69186800245}
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