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Simplifying -2k2 + -29k + 24 = 0 Reorder the terms: 24 + -29k + -2k2 = 0 Solving 24 + -29k + -2k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -12 + 14.5k + k2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 14.5k + 12 + k2 = 0 + 12 Reorder the terms: -12 + 12 + 14.5k + k2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 14.5k + k2 = 0 + 12 14.5k + k2 = 0 + 12 Combine like terms: 0 + 12 = 12 14.5k + k2 = 12 The k term is 14.5k. Take half its coefficient (7.25). Square it (52.5625) and add it to both sides. Add '52.5625' to each side of the equation. 14.5k + 52.5625 + k2 = 12 + 52.5625 Reorder the terms: 52.5625 + 14.5k + k2 = 12 + 52.5625 Combine like terms: 12 + 52.5625 = 64.5625 52.5625 + 14.5k + k2 = 64.5625 Factor a perfect square on the left side: (k + 7.25)(k + 7.25) = 64.5625 Calculate the square root of the right side: 8.03507934 Break this problem into two subproblems by setting (k + 7.25) equal to 8.03507934 and -8.03507934.Subproblem 1
k + 7.25 = 8.03507934 Simplifying k + 7.25 = 8.03507934 Reorder the terms: 7.25 + k = 8.03507934 Solving 7.25 + k = 8.03507934 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-7.25' to each side of the equation. 7.25 + -7.25 + k = 8.03507934 + -7.25 Combine like terms: 7.25 + -7.25 = 0.00 0.00 + k = 8.03507934 + -7.25 k = 8.03507934 + -7.25 Combine like terms: 8.03507934 + -7.25 = 0.78507934 k = 0.78507934 Simplifying k = 0.78507934Subproblem 2
k + 7.25 = -8.03507934 Simplifying k + 7.25 = -8.03507934 Reorder the terms: 7.25 + k = -8.03507934 Solving 7.25 + k = -8.03507934 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-7.25' to each side of the equation. 7.25 + -7.25 + k = -8.03507934 + -7.25 Combine like terms: 7.25 + -7.25 = 0.00 0.00 + k = -8.03507934 + -7.25 k = -8.03507934 + -7.25 Combine like terms: -8.03507934 + -7.25 = -15.28507934 k = -15.28507934 Simplifying k = -15.28507934Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.78507934, -15.28507934}
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