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Simplifying k2 + -31 + -2k = -4 + -2k2 + 2k Reorder the terms: -31 + -2k + k2 = -4 + -2k2 + 2k Reorder the terms: -31 + -2k + k2 = -4 + 2k + -2k2 Solving -31 + -2k + k2 = -4 + 2k + -2k2 Solving for variable 'k'. Reorder the terms: -31 + 4 + -2k + -2k + k2 + 2k2 = -4 + 2k + -2k2 + 4 + -2k + 2k2 Combine like terms: -31 + 4 = -27 -27 + -2k + -2k + k2 + 2k2 = -4 + 2k + -2k2 + 4 + -2k + 2k2 Combine like terms: -2k + -2k = -4k -27 + -4k + k2 + 2k2 = -4 + 2k + -2k2 + 4 + -2k + 2k2 Combine like terms: k2 + 2k2 = 3k2 -27 + -4k + 3k2 = -4 + 2k + -2k2 + 4 + -2k + 2k2 Reorder the terms: -27 + -4k + 3k2 = -4 + 4 + 2k + -2k + -2k2 + 2k2 Combine like terms: -4 + 4 = 0 -27 + -4k + 3k2 = 0 + 2k + -2k + -2k2 + 2k2 -27 + -4k + 3k2 = 2k + -2k + -2k2 + 2k2 Combine like terms: 2k + -2k = 0 -27 + -4k + 3k2 = 0 + -2k2 + 2k2 -27 + -4k + 3k2 = -2k2 + 2k2 Combine like terms: -2k2 + 2k2 = 0 -27 + -4k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -9 + -1.333333333k + k2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + -1.333333333k + 9 + k2 = 0 + 9 Reorder the terms: -9 + 9 + -1.333333333k + k2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + -1.333333333k + k2 = 0 + 9 -1.333333333k + k2 = 0 + 9 Combine like terms: 0 + 9 = 9 -1.333333333k + k2 = 9 The k term is -1.333333333k. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333k + 0.4444444442 + k2 = 9 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333k + k2 = 9 + 0.4444444442 Combine like terms: 9 + 0.4444444442 = 9.4444444442 0.4444444442 + -1.333333333k + k2 = 9.4444444442 Factor a perfect square on the left side: (k + -0.6666666665)(k + -0.6666666665) = 9.4444444442 Calculate the square root of the right side: 3.073181486 Break this problem into two subproblems by setting (k + -0.6666666665) equal to 3.073181486 and -3.073181486.Subproblem 1
k + -0.6666666665 = 3.073181486 Simplifying k + -0.6666666665 = 3.073181486 Reorder the terms: -0.6666666665 + k = 3.073181486 Solving -0.6666666665 + k = 3.073181486 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + k = 3.073181486 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + k = 3.073181486 + 0.6666666665 k = 3.073181486 + 0.6666666665 Combine like terms: 3.073181486 + 0.6666666665 = 3.7398481525 k = 3.7398481525 Simplifying k = 3.7398481525Subproblem 2
k + -0.6666666665 = -3.073181486 Simplifying k + -0.6666666665 = -3.073181486 Reorder the terms: -0.6666666665 + k = -3.073181486 Solving -0.6666666665 + k = -3.073181486 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + k = -3.073181486 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + k = -3.073181486 + 0.6666666665 k = -3.073181486 + 0.6666666665 Combine like terms: -3.073181486 + 0.6666666665 = -2.4065148195 k = -2.4065148195 Simplifying k = -2.4065148195Solution
The solution to the problem is based on the solutions from the subproblems. k = {3.7398481525, -2.4065148195}
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