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Simplifying 4k2 + -1k + -4 = 0 Reorder the terms: -4 + -1k + 4k2 = 0 Solving -4 + -1k + 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'. -1 + -0.25k + k2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.25k + 1 + k2 = 0 + 1 Reorder the terms: -1 + 1 + -0.25k + k2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.25k + k2 = 0 + 1 -0.25k + k2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.25k + k2 = 1 The k term is -0.25k. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25k + 0.015625 + k2 = 1 + 0.015625 Reorder the terms: 0.015625 + -0.25k + k2 = 1 + 0.015625 Combine like terms: 1 + 0.015625 = 1.015625 0.015625 + -0.25k + k2 = 1.015625 Factor a perfect square on the left side: (k + -0.125)(k + -0.125) = 1.015625 Calculate the square root of the right side: 1.007782219 Break this problem into two subproblems by setting (k + -0.125) equal to 1.007782219 and -1.007782219.Subproblem 1
k + -0.125 = 1.007782219 Simplifying k + -0.125 = 1.007782219 Reorder the terms: -0.125 + k = 1.007782219 Solving -0.125 + k = 1.007782219 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + k = 1.007782219 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + k = 1.007782219 + 0.125 k = 1.007782219 + 0.125 Combine like terms: 1.007782219 + 0.125 = 1.132782219 k = 1.132782219 Simplifying k = 1.132782219Subproblem 2
k + -0.125 = -1.007782219 Simplifying k + -0.125 = -1.007782219 Reorder the terms: -0.125 + k = -1.007782219 Solving -0.125 + k = -1.007782219 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + k = -1.007782219 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + k = -1.007782219 + 0.125 k = -1.007782219 + 0.125 Combine like terms: -1.007782219 + 0.125 = -0.882782219 k = -0.882782219 Simplifying k = -0.882782219Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.132782219, -0.882782219}
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