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Simplifying 3k2 + 5k + -24 = 0 Reorder the terms: -24 + 5k + 3k2 = 0 Solving -24 + 5k + 3k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -8 + 1.666666667k + k2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + 1.666666667k + 8 + k2 = 0 + 8 Reorder the terms: -8 + 8 + 1.666666667k + k2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 1.666666667k + k2 = 0 + 8 1.666666667k + k2 = 0 + 8 Combine like terms: 0 + 8 = 8 1.666666667k + k2 = 8 The k term is 1.666666667k. Take half its coefficient (0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. 1.666666667k + 0.6944444447 + k2 = 8 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667k + k2 = 8 + 0.6944444447 Combine like terms: 8 + 0.6944444447 = 8.6944444447 0.6944444447 + 1.666666667k + k2 = 8.6944444447 Factor a perfect square on the left side: (k + 0.8333333335)(k + 0.8333333335) = 8.6944444447 Calculate the square root of the right side: 2.948634336 Break this problem into two subproblems by setting (k + 0.8333333335) equal to 2.948634336 and -2.948634336.Subproblem 1
k + 0.8333333335 = 2.948634336 Simplifying k + 0.8333333335 = 2.948634336 Reorder the terms: 0.8333333335 + k = 2.948634336 Solving 0.8333333335 + k = 2.948634336 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + k = 2.948634336 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + k = 2.948634336 + -0.8333333335 k = 2.948634336 + -0.8333333335 Combine like terms: 2.948634336 + -0.8333333335 = 2.1153010025 k = 2.1153010025 Simplifying k = 2.1153010025Subproblem 2
k + 0.8333333335 = -2.948634336 Simplifying k + 0.8333333335 = -2.948634336 Reorder the terms: 0.8333333335 + k = -2.948634336 Solving 0.8333333335 + k = -2.948634336 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + k = -2.948634336 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + k = -2.948634336 + -0.8333333335 k = -2.948634336 + -0.8333333335 Combine like terms: -2.948634336 + -0.8333333335 = -3.7819676695 k = -3.7819676695 Simplifying k = -3.7819676695Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.1153010025, -3.7819676695}
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