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Simplifying 8k2 + 24k + -66 = 0 Reorder the terms: -66 + 24k + 8k2 = 0 Solving -66 + 24k + 8k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2'. 2(-33 + 12k + 4k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-33 + 12k + 4k2)' equal to zero and attempt to solve: Simplifying -33 + 12k + 4k2 = 0 Solving -33 + 12k + 4k2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -8.25 + 3k + k2 = 0 Move the constant term to the right: Add '8.25' to each side of the equation. -8.25 + 3k + 8.25 + k2 = 0 + 8.25 Reorder the terms: -8.25 + 8.25 + 3k + k2 = 0 + 8.25 Combine like terms: -8.25 + 8.25 = 0.00 0.00 + 3k + k2 = 0 + 8.25 3k + k2 = 0 + 8.25 Combine like terms: 0 + 8.25 = 8.25 3k + k2 = 8.25 The k term is 3k. 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. 3k + 2.25 + k2 = 8.25 + 2.25 Reorder the terms: 2.25 + 3k + k2 = 8.25 + 2.25 Combine like terms: 8.25 + 2.25 = 10.5 2.25 + 3k + k2 = 10.5 Factor a perfect square on the left side: (k + 1.5)(k + 1.5) = 10.5 Calculate the square root of the right side: 3.240370349 Break this problem into two subproblems by setting (k + 1.5) equal to 3.240370349 and -3.240370349.Subproblem 1
k + 1.5 = 3.240370349 Simplifying k + 1.5 = 3.240370349 Reorder the terms: 1.5 + k = 3.240370349 Solving 1.5 + k = 3.240370349 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + k = 3.240370349 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + k = 3.240370349 + -1.5 k = 3.240370349 + -1.5 Combine like terms: 3.240370349 + -1.5 = 1.740370349 k = 1.740370349 Simplifying k = 1.740370349Subproblem 2
k + 1.5 = -3.240370349 Simplifying k + 1.5 = -3.240370349 Reorder the terms: 1.5 + k = -3.240370349 Solving 1.5 + k = -3.240370349 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + k = -3.240370349 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + k = -3.240370349 + -1.5 k = -3.240370349 + -1.5 Combine like terms: -3.240370349 + -1.5 = -4.740370349 k = -4.740370349 Simplifying k = -4.740370349Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.740370349, -4.740370349}Solution
k = {1.740370349, -4.740370349}
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