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Simplifying 9 + 2k(7 + -2k) * 5 = 0 Reorder the terms for easier multiplication: 9 + 2 * 5k(7 + -2k) = 0 Multiply 2 * 5 9 + 10k(7 + -2k) = 0 9 + (7 * 10k + -2k * 10k) = 0 9 + (70k + -20k2) = 0 Solving 9 + 70k + -20k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by -20 the coefficient of the squared term: Divide each side by '-20'. -0.45 + -3.5k + k2 = 0.0 Move the constant term to the right: Add '0.45' to each side of the equation. -0.45 + -3.5k + 0.45 + k2 = 0.0 + 0.45 Reorder the terms: -0.45 + 0.45 + -3.5k + k2 = 0.0 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + -3.5k + k2 = 0.0 + 0.45 -3.5k + k2 = 0.0 + 0.45 Combine like terms: 0.0 + 0.45 = 0.45 -3.5k + k2 = 0.45 The k term is -3.5k. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5k + 3.0625 + k2 = 0.45 + 3.0625 Reorder the terms: 3.0625 + -3.5k + k2 = 0.45 + 3.0625 Combine like terms: 0.45 + 3.0625 = 3.5125 3.0625 + -3.5k + k2 = 3.5125 Factor a perfect square on the left side: (k + -1.75)(k + -1.75) = 3.5125 Calculate the square root of the right side: 1.874166481 Break this problem into two subproblems by setting (k + -1.75) equal to 1.874166481 and -1.874166481.Subproblem 1
k + -1.75 = 1.874166481 Simplifying k + -1.75 = 1.874166481 Reorder the terms: -1.75 + k = 1.874166481 Solving -1.75 + k = 1.874166481 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + k = 1.874166481 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + k = 1.874166481 + 1.75 k = 1.874166481 + 1.75 Combine like terms: 1.874166481 + 1.75 = 3.624166481 k = 3.624166481 Simplifying k = 3.624166481Subproblem 2
k + -1.75 = -1.874166481 Simplifying k + -1.75 = -1.874166481 Reorder the terms: -1.75 + k = -1.874166481 Solving -1.75 + k = -1.874166481 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + k = -1.874166481 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + k = -1.874166481 + 1.75 k = -1.874166481 + 1.75 Combine like terms: -1.874166481 + 1.75 = -0.124166481 k = -0.124166481 Simplifying k = -0.124166481Solution
The solution to the problem is based on the solutions from the subproblems. k = {3.624166481, -0.124166481}
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