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Simplifying 4k2 + -5k + -2 = 0 Reorder the terms: -2 + -5k + 4k2 = 0 Solving -2 + -5k + 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'. -0.5 + -1.25k + k2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1.25k + 0.5 + k2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1.25k + k2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1.25k + k2 = 0 + 0.5 -1.25k + k2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1.25k + k2 = 0.5 The k term is -1.25k. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25k + 0.390625 + k2 = 0.5 + 0.390625 Reorder the terms: 0.390625 + -1.25k + k2 = 0.5 + 0.390625 Combine like terms: 0.5 + 0.390625 = 0.890625 0.390625 + -1.25k + k2 = 0.890625 Factor a perfect square on the left side: (k + -0.625)(k + -0.625) = 0.890625 Calculate the square root of the right side: 0.943729304 Break this problem into two subproblems by setting (k + -0.625) equal to 0.943729304 and -0.943729304.Subproblem 1
k + -0.625 = 0.943729304 Simplifying k + -0.625 = 0.943729304 Reorder the terms: -0.625 + k = 0.943729304 Solving -0.625 + k = 0.943729304 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + k = 0.943729304 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + k = 0.943729304 + 0.625 k = 0.943729304 + 0.625 Combine like terms: 0.943729304 + 0.625 = 1.568729304 k = 1.568729304 Simplifying k = 1.568729304Subproblem 2
k + -0.625 = -0.943729304 Simplifying k + -0.625 = -0.943729304 Reorder the terms: -0.625 + k = -0.943729304 Solving -0.625 + k = -0.943729304 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + k = -0.943729304 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + k = -0.943729304 + 0.625 k = -0.943729304 + 0.625 Combine like terms: -0.943729304 + 0.625 = -0.318729304 k = -0.318729304 Simplifying k = -0.318729304Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.568729304, -0.318729304}
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