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Simplifying (4k2 + 8k + 2) + -1(2k + 3) = 0 Reorder the terms: (2 + 8k + 4k2) + -1(2k + 3) = 0 Remove parenthesis around (2 + 8k + 4k2) 2 + 8k + 4k2 + -1(2k + 3) = 0 Reorder the terms: 2 + 8k + 4k2 + -1(3 + 2k) = 0 2 + 8k + 4k2 + (3 * -1 + 2k * -1) = 0 2 + 8k + 4k2 + (-3 + -2k) = 0 Reorder the terms: 2 + -3 + 8k + -2k + 4k2 = 0 Combine like terms: 2 + -3 = -1 -1 + 8k + -2k + 4k2 = 0 Combine like terms: 8k + -2k = 6k -1 + 6k + 4k2 = 0 Solving -1 + 6k + 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 + 1.5k + k2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 1.5k + 0.25 + k2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 1.5k + k2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 1.5k + k2 = 0 + 0.25 1.5k + k2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 1.5k + k2 = 0.25 The k term is 1.5k. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5k + 0.5625 + k2 = 0.25 + 0.5625 Reorder the terms: 0.5625 + 1.5k + k2 = 0.25 + 0.5625 Combine like terms: 0.25 + 0.5625 = 0.8125 0.5625 + 1.5k + k2 = 0.8125 Factor a perfect square on the left side: (k + 0.75)(k + 0.75) = 0.8125 Calculate the square root of the right side: 0.901387819 Break this problem into two subproblems by setting (k + 0.75) equal to 0.901387819 and -0.901387819.Subproblem 1
k + 0.75 = 0.901387819 Simplifying k + 0.75 = 0.901387819 Reorder the terms: 0.75 + k = 0.901387819 Solving 0.75 + k = 0.901387819 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + k = 0.901387819 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + k = 0.901387819 + -0.75 k = 0.901387819 + -0.75 Combine like terms: 0.901387819 + -0.75 = 0.151387819 k = 0.151387819 Simplifying k = 0.151387819Subproblem 2
k + 0.75 = -0.901387819 Simplifying k + 0.75 = -0.901387819 Reorder the terms: 0.75 + k = -0.901387819 Solving 0.75 + k = -0.901387819 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + k = -0.901387819 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + k = -0.901387819 + -0.75 k = -0.901387819 + -0.75 Combine like terms: -0.901387819 + -0.75 = -1.651387819 k = -1.651387819 Simplifying k = -1.651387819Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.151387819, -1.651387819}
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