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Simplifying 4k2 + -14k + -1 = 0 Reorder the terms: -1 + -14k + 4k2 = 0 Solving -1 + -14k + 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'. -0.25 + -3.5k + k2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -3.5k + 0.25 + k2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -3.5k + k2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -3.5k + k2 = 0 + 0.25 -3.5k + k2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -3.5k + k2 = 0.25 The k term is -3.5k. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5k + 3.0625 + k2 = 0.25 + 3.0625 Reorder the terms: 3.0625 + -3.5k + k2 = 0.25 + 3.0625 Combine like terms: 0.25 + 3.0625 = 3.3125 3.0625 + -3.5k + k2 = 3.3125 Factor a perfect square on the left side: (k + -1.75)(k + -1.75) = 3.3125 Calculate the square root of the right side: 1.820027472 Break this problem into two subproblems by setting (k + -1.75) equal to 1.820027472 and -1.820027472.Subproblem 1
k + -1.75 = 1.820027472 Simplifying k + -1.75 = 1.820027472 Reorder the terms: -1.75 + k = 1.820027472 Solving -1.75 + k = 1.820027472 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + k = 1.820027472 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + k = 1.820027472 + 1.75 k = 1.820027472 + 1.75 Combine like terms: 1.820027472 + 1.75 = 3.570027472 k = 3.570027472 Simplifying k = 3.570027472Subproblem 2
k + -1.75 = -1.820027472 Simplifying k + -1.75 = -1.820027472 Reorder the terms: -1.75 + k = -1.820027472 Solving -1.75 + k = -1.820027472 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + k = -1.820027472 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + k = -1.820027472 + 1.75 k = -1.820027472 + 1.75 Combine like terms: -1.820027472 + 1.75 = -0.070027472 k = -0.070027472 Simplifying k = -0.070027472Solution
The solution to the problem is based on the solutions from the subproblems. k = {3.570027472, -0.070027472}
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