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Simplifying -1k(-8k) + -15 + 3k + -2 = 0 Remove parenthesis around (-8k) -1k * -8k + -15 + 3k + -2 = 0 Reorder the terms for easier multiplication: -1 * -8k * k + -15 + 3k + -2 = 0 Multiply -1 * -8 8k * k + -15 + 3k + -2 = 0 Multiply k * k 8k2 + -15 + 3k + -2 = 0 Reorder the terms: -15 + -2 + 3k + 8k2 = 0 Combine like terms: -15 + -2 = -17 -17 + 3k + 8k2 = 0 Solving -17 + 3k + 8k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.125 + 0.375k + k2 = 0 Move the constant term to the right: Add '2.125' to each side of the equation. -2.125 + 0.375k + 2.125 + k2 = 0 + 2.125 Reorder the terms: -2.125 + 2.125 + 0.375k + k2 = 0 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + 0.375k + k2 = 0 + 2.125 0.375k + k2 = 0 + 2.125 Combine like terms: 0 + 2.125 = 2.125 0.375k + k2 = 2.125 The k term is 0.375k. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375k + 0.03515625 + k2 = 2.125 + 0.03515625 Reorder the terms: 0.03515625 + 0.375k + k2 = 2.125 + 0.03515625 Combine like terms: 2.125 + 0.03515625 = 2.16015625 0.03515625 + 0.375k + k2 = 2.16015625 Factor a perfect square on the left side: (k + 0.1875)(k + 0.1875) = 2.16015625 Calculate the square root of the right side: 1.469747002 Break this problem into two subproblems by setting (k + 0.1875) equal to 1.469747002 and -1.469747002.Subproblem 1
k + 0.1875 = 1.469747002 Simplifying k + 0.1875 = 1.469747002 Reorder the terms: 0.1875 + k = 1.469747002 Solving 0.1875 + k = 1.469747002 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + k = 1.469747002 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + k = 1.469747002 + -0.1875 k = 1.469747002 + -0.1875 Combine like terms: 1.469747002 + -0.1875 = 1.282247002 k = 1.282247002 Simplifying k = 1.282247002Subproblem 2
k + 0.1875 = -1.469747002 Simplifying k + 0.1875 = -1.469747002 Reorder the terms: 0.1875 + k = -1.469747002 Solving 0.1875 + k = -1.469747002 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + k = -1.469747002 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + k = -1.469747002 + -0.1875 k = -1.469747002 + -0.1875 Combine like terms: -1.469747002 + -0.1875 = -1.657247002 k = -1.657247002 Simplifying k = -1.657247002Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.282247002, -1.657247002}
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