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Simplifying 5k2 + -10k + 3 = 0 Reorder the terms: 3 + -10k + 5k2 = 0 Solving 3 + -10k + 5k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.6 + -2k + k2 = 0 Move the constant term to the right: Add '-0.6' to each side of the equation. 0.6 + -2k + -0.6 + k2 = 0 + -0.6 Reorder the terms: 0.6 + -0.6 + -2k + k2 = 0 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + -2k + k2 = 0 + -0.6 -2k + k2 = 0 + -0.6 Combine like terms: 0 + -0.6 = -0.6 -2k + k2 = -0.6 The k term is -2k. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2k + 1 + k2 = -0.6 + 1 Reorder the terms: 1 + -2k + k2 = -0.6 + 1 Combine like terms: -0.6 + 1 = 0.4 1 + -2k + k2 = 0.4 Factor a perfect square on the left side: (k + -1)(k + -1) = 0.4 Calculate the square root of the right side: 0.632455532 Break this problem into two subproblems by setting (k + -1) equal to 0.632455532 and -0.632455532.Subproblem 1
k + -1 = 0.632455532 Simplifying k + -1 = 0.632455532 Reorder the terms: -1 + k = 0.632455532 Solving -1 + k = 0.632455532 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + k = 0.632455532 + 1 Combine like terms: -1 + 1 = 0 0 + k = 0.632455532 + 1 k = 0.632455532 + 1 Combine like terms: 0.632455532 + 1 = 1.632455532 k = 1.632455532 Simplifying k = 1.632455532Subproblem 2
k + -1 = -0.632455532 Simplifying k + -1 = -0.632455532 Reorder the terms: -1 + k = -0.632455532 Solving -1 + k = -0.632455532 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + k = -0.632455532 + 1 Combine like terms: -1 + 1 = 0 0 + k = -0.632455532 + 1 k = -0.632455532 + 1 Combine like terms: -0.632455532 + 1 = 0.367544468 k = 0.367544468 Simplifying k = 0.367544468Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.632455532, 0.367544468}
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