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Simplifying -2k(k + -5) + 3 = 0 Reorder the terms: -2k(-5 + k) + 3 = 0 (-5 * -2k + k * -2k) + 3 = 0 (10k + -2k2) + 3 = 0 Reorder the terms: 3 + 10k + -2k2 = 0 Solving 3 + 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'. -1.5 + -5k + k2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -5k + 1.5 + k2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -5k + k2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -5k + k2 = 0 + 1.5 -5k + k2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -5k + k2 = 1.5 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 = 1.5 + 6.25 Reorder the terms: 6.25 + -5k + k2 = 1.5 + 6.25 Combine like terms: 1.5 + 6.25 = 7.75 6.25 + -5k + k2 = 7.75 Factor a perfect square on the left side: (k + -2.5)(k + -2.5) = 7.75 Calculate the square root of the right side: 2.783882181 Break this problem into two subproblems by setting (k + -2.5) equal to 2.783882181 and -2.783882181.Subproblem 1
k + -2.5 = 2.783882181 Simplifying k + -2.5 = 2.783882181 Reorder the terms: -2.5 + k = 2.783882181 Solving -2.5 + k = 2.783882181 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 = 2.783882181 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + k = 2.783882181 + 2.5 k = 2.783882181 + 2.5 Combine like terms: 2.783882181 + 2.5 = 5.283882181 k = 5.283882181 Simplifying k = 5.283882181Subproblem 2
k + -2.5 = -2.783882181 Simplifying k + -2.5 = -2.783882181 Reorder the terms: -2.5 + k = -2.783882181 Solving -2.5 + k = -2.783882181 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 = -2.783882181 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + k = -2.783882181 + 2.5 k = -2.783882181 + 2.5 Combine like terms: -2.783882181 + 2.5 = -0.283882181 k = -0.283882181 Simplifying k = -0.283882181Solution
The solution to the problem is based on the solutions from the subproblems. k = {5.283882181, -0.283882181}
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