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Simplifying 4 * 2 + 5k = 2k2 + 2k + 7 Multiply 4 * 2 8 + 5k = 2k2 + 2k + 7 Reorder the terms: 8 + 5k = 7 + 2k + 2k2 Solving 8 + 5k = 7 + 2k + 2k2 Solving for variable 'k'. Reorder the terms: 8 + -7 + 5k + -2k + -2k2 = 7 + 2k + 2k2 + -7 + -2k + -2k2 Combine like terms: 8 + -7 = 1 1 + 5k + -2k + -2k2 = 7 + 2k + 2k2 + -7 + -2k + -2k2 Combine like terms: 5k + -2k = 3k 1 + 3k + -2k2 = 7 + 2k + 2k2 + -7 + -2k + -2k2 Reorder the terms: 1 + 3k + -2k2 = 7 + -7 + 2k + -2k + 2k2 + -2k2 Combine like terms: 7 + -7 = 0 1 + 3k + -2k2 = 0 + 2k + -2k + 2k2 + -2k2 1 + 3k + -2k2 = 2k + -2k + 2k2 + -2k2 Combine like terms: 2k + -2k = 0 1 + 3k + -2k2 = 0 + 2k2 + -2k2 1 + 3k + -2k2 = 2k2 + -2k2 Combine like terms: 2k2 + -2k2 = 0 1 + 3k + -2k2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.5 + -1.5k + k2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1.5k + 0.5 + k2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1.5k + k2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1.5k + k2 = 0 + 0.5 -1.5k + k2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1.5k + k2 = 0.5 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 = 0.5 + 0.5625 Reorder the terms: 0.5625 + -1.5k + k2 = 0.5 + 0.5625 Combine like terms: 0.5 + 0.5625 = 1.0625 0.5625 + -1.5k + k2 = 1.0625 Factor a perfect square on the left side: (k + -0.75)(k + -0.75) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (k + -0.75) equal to 1.030776406 and -1.030776406.Subproblem 1
k + -0.75 = 1.030776406 Simplifying k + -0.75 = 1.030776406 Reorder the terms: -0.75 + k = 1.030776406 Solving -0.75 + k = 1.030776406 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 = 1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + k = 1.030776406 + 0.75 k = 1.030776406 + 0.75 Combine like terms: 1.030776406 + 0.75 = 1.780776406 k = 1.780776406 Simplifying k = 1.780776406Subproblem 2
k + -0.75 = -1.030776406 Simplifying k + -0.75 = -1.030776406 Reorder the terms: -0.75 + k = -1.030776406 Solving -0.75 + k = -1.030776406 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 = -1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + k = -1.030776406 + 0.75 k = -1.030776406 + 0.75 Combine like terms: -1.030776406 + 0.75 = -0.280776406 k = -0.280776406 Simplifying k = -0.280776406Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.780776406, -0.280776406}
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