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Simplifying 2k2 + 10k + 7 = 0 Reorder the terms: 7 + 10k + 2k2 = 0 Solving 7 + 10k + 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'. 3.5 + 5k + k2 = 0 Move the constant term to the right: Add '-3.5' to each side of the equation. 3.5 + 5k + -3.5 + k2 = 0 + -3.5 Reorder the terms: 3.5 + -3.5 + 5k + k2 = 0 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + 5k + k2 = 0 + -3.5 5k + k2 = 0 + -3.5 Combine like terms: 0 + -3.5 = -3.5 5k + k2 = -3.5 The k term is 5k. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5k + 6.25 + k2 = -3.5 + 6.25 Reorder the terms: 6.25 + 5k + k2 = -3.5 + 6.25 Combine like terms: -3.5 + 6.25 = 2.75 6.25 + 5k + k2 = 2.75 Factor a perfect square on the left side: (k + 2.5)(k + 2.5) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (k + 2.5) equal to 1.658312395 and -1.658312395.Subproblem 1
k + 2.5 = 1.658312395 Simplifying k + 2.5 = 1.658312395 Reorder the terms: 2.5 + k = 1.658312395 Solving 2.5 + k = 1.658312395 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + k = 1.658312395 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + k = 1.658312395 + -2.5 k = 1.658312395 + -2.5 Combine like terms: 1.658312395 + -2.5 = -0.841687605 k = -0.841687605 Simplifying k = -0.841687605Subproblem 2
k + 2.5 = -1.658312395 Simplifying k + 2.5 = -1.658312395 Reorder the terms: 2.5 + k = -1.658312395 Solving 2.5 + k = -1.658312395 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + k = -1.658312395 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + k = -1.658312395 + -2.5 k = -1.658312395 + -2.5 Combine like terms: -1.658312395 + -2.5 = -4.158312395 k = -4.158312395 Simplifying k = -4.158312395Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.841687605, -4.158312395}
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