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Simplifying 5k2 + -45k + 8 = 0 Reorder the terms: 8 + -45k + 5k2 = 0 Solving 8 + -45k + 5k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 1.6 + -9k + k2 = 0 Move the constant term to the right: Add '-1.6' to each side of the equation. 1.6 + -9k + -1.6 + k2 = 0 + -1.6 Reorder the terms: 1.6 + -1.6 + -9k + k2 = 0 + -1.6 Combine like terms: 1.6 + -1.6 = 0.0 0.0 + -9k + k2 = 0 + -1.6 -9k + k2 = 0 + -1.6 Combine like terms: 0 + -1.6 = -1.6 -9k + k2 = -1.6 The k term is -9k. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9k + 20.25 + k2 = -1.6 + 20.25 Reorder the terms: 20.25 + -9k + k2 = -1.6 + 20.25 Combine like terms: -1.6 + 20.25 = 18.65 20.25 + -9k + k2 = 18.65 Factor a perfect square on the left side: (k + -4.5)(k + -4.5) = 18.65 Calculate the square root of the right side: 4.318564576 Break this problem into two subproblems by setting (k + -4.5) equal to 4.318564576 and -4.318564576.Subproblem 1
k + -4.5 = 4.318564576 Simplifying k + -4.5 = 4.318564576 Reorder the terms: -4.5 + k = 4.318564576 Solving -4.5 + k = 4.318564576 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + k = 4.318564576 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + k = 4.318564576 + 4.5 k = 4.318564576 + 4.5 Combine like terms: 4.318564576 + 4.5 = 8.818564576 k = 8.818564576 Simplifying k = 8.818564576Subproblem 2
k + -4.5 = -4.318564576 Simplifying k + -4.5 = -4.318564576 Reorder the terms: -4.5 + k = -4.318564576 Solving -4.5 + k = -4.318564576 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + k = -4.318564576 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + k = -4.318564576 + 4.5 k = -4.318564576 + 4.5 Combine like terms: -4.318564576 + 4.5 = 0.181435424 k = 0.181435424 Simplifying k = 0.181435424Solution
The solution to the problem is based on the solutions from the subproblems. k = {8.818564576, 0.181435424}
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