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Simplifying -4k(k + -2) + 14 = 3k + -20 Reorder the terms: -4k(-2 + k) + 14 = 3k + -20 (-2 * -4k + k * -4k) + 14 = 3k + -20 (8k + -4k2) + 14 = 3k + -20 Reorder the terms: 14 + 8k + -4k2 = 3k + -20 Reorder the terms: 14 + 8k + -4k2 = -20 + 3k Solving 14 + 8k + -4k2 = -20 + 3k Solving for variable 'k'. Reorder the terms: 14 + 20 + 8k + -3k + -4k2 = -20 + 3k + 20 + -3k Combine like terms: 14 + 20 = 34 34 + 8k + -3k + -4k2 = -20 + 3k + 20 + -3k Combine like terms: 8k + -3k = 5k 34 + 5k + -4k2 = -20 + 3k + 20 + -3k Reorder the terms: 34 + 5k + -4k2 = -20 + 20 + 3k + -3k Combine like terms: -20 + 20 = 0 34 + 5k + -4k2 = 0 + 3k + -3k 34 + 5k + -4k2 = 3k + -3k Combine like terms: 3k + -3k = 0 34 + 5k + -4k2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -8.5 + -1.25k + k2 = 0 Move the constant term to the right: Add '8.5' to each side of the equation. -8.5 + -1.25k + 8.5 + k2 = 0 + 8.5 Reorder the terms: -8.5 + 8.5 + -1.25k + k2 = 0 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + -1.25k + k2 = 0 + 8.5 -1.25k + k2 = 0 + 8.5 Combine like terms: 0 + 8.5 = 8.5 -1.25k + k2 = 8.5 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 = 8.5 + 0.390625 Reorder the terms: 0.390625 + -1.25k + k2 = 8.5 + 0.390625 Combine like terms: 8.5 + 0.390625 = 8.890625 0.390625 + -1.25k + k2 = 8.890625 Factor a perfect square on the left side: (k + -0.625)(k + -0.625) = 8.890625 Calculate the square root of the right side: 2.98171511 Break this problem into two subproblems by setting (k + -0.625) equal to 2.98171511 and -2.98171511.Subproblem 1
k + -0.625 = 2.98171511 Simplifying k + -0.625 = 2.98171511 Reorder the terms: -0.625 + k = 2.98171511 Solving -0.625 + k = 2.98171511 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 = 2.98171511 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + k = 2.98171511 + 0.625 k = 2.98171511 + 0.625 Combine like terms: 2.98171511 + 0.625 = 3.60671511 k = 3.60671511 Simplifying k = 3.60671511Subproblem 2
k + -0.625 = -2.98171511 Simplifying k + -0.625 = -2.98171511 Reorder the terms: -0.625 + k = -2.98171511 Solving -0.625 + k = -2.98171511 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 = -2.98171511 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + k = -2.98171511 + 0.625 k = -2.98171511 + 0.625 Combine like terms: -2.98171511 + 0.625 = -2.35671511 k = -2.35671511 Simplifying k = -2.35671511Solution
The solution to the problem is based on the solutions from the subproblems. k = {3.60671511, -2.35671511}
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