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Simplifying 2k2 + 5k + -2 = 0 Reorder the terms: -2 + 5k + 2k2 = 0 Solving -2 + 5k + 2k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1 + 2.5k + k2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 2.5k + 1 + k2 = 0 + 1 Reorder the terms: -1 + 1 + 2.5k + k2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2.5k + k2 = 0 + 1 2.5k + k2 = 0 + 1 Combine like terms: 0 + 1 = 1 2.5k + k2 = 1 The k term is 2.5k. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5k + 1.5625 + k2 = 1 + 1.5625 Reorder the terms: 1.5625 + 2.5k + k2 = 1 + 1.5625 Combine like terms: 1 + 1.5625 = 2.5625 1.5625 + 2.5k + k2 = 2.5625 Factor a perfect square on the left side: (k + 1.25)(k + 1.25) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (k + 1.25) equal to 1.600781059 and -1.600781059.Subproblem 1
k + 1.25 = 1.600781059 Simplifying k + 1.25 = 1.600781059 Reorder the terms: 1.25 + k = 1.600781059 Solving 1.25 + k = 1.600781059 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + k = 1.600781059 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + k = 1.600781059 + -1.25 k = 1.600781059 + -1.25 Combine like terms: 1.600781059 + -1.25 = 0.350781059 k = 0.350781059 Simplifying k = 0.350781059Subproblem 2
k + 1.25 = -1.600781059 Simplifying k + 1.25 = -1.600781059 Reorder the terms: 1.25 + k = -1.600781059 Solving 1.25 + k = -1.600781059 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + k = -1.600781059 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + k = -1.600781059 + -1.25 k = -1.600781059 + -1.25 Combine like terms: -1.600781059 + -1.25 = -2.850781059 k = -2.850781059 Simplifying k = -2.850781059Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.350781059, -2.850781059}
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