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Simplifying 8k2 + -18k + -10 = 0 Reorder the terms: -10 + -18k + 8k2 = 0 Solving -10 + -18k + 8k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + -9k + 4k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + -9k + 4k2)' equal to zero and attempt to solve: Simplifying -5 + -9k + 4k2 = 0 Solving -5 + -9k + 4k2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.25 + -2.25k + k2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -2.25k + 1.25 + k2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -2.25k + k2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -2.25k + k2 = 0 + 1.25 -2.25k + k2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -2.25k + k2 = 1.25 The k term is -2.25k. Take half its coefficient (-1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. -2.25k + 1.265625 + k2 = 1.25 + 1.265625 Reorder the terms: 1.265625 + -2.25k + k2 = 1.25 + 1.265625 Combine like terms: 1.25 + 1.265625 = 2.515625 1.265625 + -2.25k + k2 = 2.515625 Factor a perfect square on the left side: (k + -1.125)(k + -1.125) = 2.515625 Calculate the square root of the right side: 1.586072193 Break this problem into two subproblems by setting (k + -1.125) equal to 1.586072193 and -1.586072193.Subproblem 1
k + -1.125 = 1.586072193 Simplifying k + -1.125 = 1.586072193 Reorder the terms: -1.125 + k = 1.586072193 Solving -1.125 + k = 1.586072193 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + k = 1.586072193 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + k = 1.586072193 + 1.125 k = 1.586072193 + 1.125 Combine like terms: 1.586072193 + 1.125 = 2.711072193 k = 2.711072193 Simplifying k = 2.711072193Subproblem 2
k + -1.125 = -1.586072193 Simplifying k + -1.125 = -1.586072193 Reorder the terms: -1.125 + k = -1.586072193 Solving -1.125 + k = -1.586072193 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + k = -1.586072193 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + k = -1.586072193 + 1.125 k = -1.586072193 + 1.125 Combine like terms: -1.586072193 + 1.125 = -0.461072193 k = -0.461072193 Simplifying k = -0.461072193Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.711072193, -0.461072193}Solution
k = {2.711072193, -0.461072193}
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