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Simplifying -4k2 + -8k + 9 = 0 Reorder the terms: 9 + -8k + -4k2 = 0 Solving 9 + -8k + -4k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -2.25 + 2k + k2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + 2k + 2.25 + k2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + 2k + k2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + 2k + k2 = 0 + 2.25 2k + k2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 2k + k2 = 2.25 The k term is 2k. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2k + 1 + k2 = 2.25 + 1 Reorder the terms: 1 + 2k + k2 = 2.25 + 1 Combine like terms: 2.25 + 1 = 3.25 1 + 2k + k2 = 3.25 Factor a perfect square on the left side: (k + 1)(k + 1) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (k + 1) equal to 1.802775638 and -1.802775638.Subproblem 1
k + 1 = 1.802775638 Simplifying k + 1 = 1.802775638 Reorder the terms: 1 + k = 1.802775638 Solving 1 + k = 1.802775638 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = 1.802775638 + -1 Combine like terms: 1 + -1 = 0 0 + k = 1.802775638 + -1 k = 1.802775638 + -1 Combine like terms: 1.802775638 + -1 = 0.802775638 k = 0.802775638 Simplifying k = 0.802775638Subproblem 2
k + 1 = -1.802775638 Simplifying k + 1 = -1.802775638 Reorder the terms: 1 + k = -1.802775638 Solving 1 + k = -1.802775638 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = -1.802775638 + -1 Combine like terms: 1 + -1 = 0 0 + k = -1.802775638 + -1 k = -1.802775638 + -1 Combine like terms: -1.802775638 + -1 = -2.802775638 k = -2.802775638 Simplifying k = -2.802775638Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.802775638, -2.802775638}
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