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Simplifying (2k + 4)(2k + 9) = 3k(3k) + 11k Reorder the terms: (4 + 2k)(2k + 9) = 3k(3k) + 11k Reorder the terms: (4 + 2k)(9 + 2k) = 3k(3k) + 11k Multiply (4 + 2k) * (9 + 2k) (4(9 + 2k) + 2k * (9 + 2k)) = 3k(3k) + 11k ((9 * 4 + 2k * 4) + 2k * (9 + 2k)) = 3k(3k) + 11k ((36 + 8k) + 2k * (9 + 2k)) = 3k(3k) + 11k (36 + 8k + (9 * 2k + 2k * 2k)) = 3k(3k) + 11k (36 + 8k + (18k + 4k2)) = 3k(3k) + 11k Combine like terms: 8k + 18k = 26k (36 + 26k + 4k2) = 3k(3k) + 11k Remove parenthesis around (3k) 36 + 26k + 4k2 = 3k * 3k + 11k Reorder the terms for easier multiplication: 36 + 26k + 4k2 = 3 * 3k * k + 11k Multiply 3 * 3 36 + 26k + 4k2 = 9k * k + 11k Multiply k * k 36 + 26k + 4k2 = 9k2 + 11k Reorder the terms: 36 + 26k + 4k2 = 11k + 9k2 Solving 36 + 26k + 4k2 = 11k + 9k2 Solving for variable 'k'. Reorder the terms: 36 + 26k + -11k + 4k2 + -9k2 = 11k + 9k2 + -11k + -9k2 Combine like terms: 26k + -11k = 15k 36 + 15k + 4k2 + -9k2 = 11k + 9k2 + -11k + -9k2 Combine like terms: 4k2 + -9k2 = -5k2 36 + 15k + -5k2 = 11k + 9k2 + -11k + -9k2 Reorder the terms: 36 + 15k + -5k2 = 11k + -11k + 9k2 + -9k2 Combine like terms: 11k + -11k = 0 36 + 15k + -5k2 = 0 + 9k2 + -9k2 36 + 15k + -5k2 = 9k2 + -9k2 Combine like terms: 9k2 + -9k2 = 0 36 + 15k + -5k2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -7.2 + -3k + k2 = 0 Move the constant term to the right: Add '7.2' to each side of the equation. -7.2 + -3k + 7.2 + k2 = 0 + 7.2 Reorder the terms: -7.2 + 7.2 + -3k + k2 = 0 + 7.2 Combine like terms: -7.2 + 7.2 = 0.0 0.0 + -3k + k2 = 0 + 7.2 -3k + k2 = 0 + 7.2 Combine like terms: 0 + 7.2 = 7.2 -3k + k2 = 7.2 The k term is -3k. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3k + 2.25 + k2 = 7.2 + 2.25 Reorder the terms: 2.25 + -3k + k2 = 7.2 + 2.25 Combine like terms: 7.2 + 2.25 = 9.45 2.25 + -3k + k2 = 9.45 Factor a perfect square on the left side: (k + -1.5)(k + -1.5) = 9.45 Calculate the square root of the right side: 3.07408523 Break this problem into two subproblems by setting (k + -1.5) equal to 3.07408523 and -3.07408523.Subproblem 1
k + -1.5 = 3.07408523 Simplifying k + -1.5 = 3.07408523 Reorder the terms: -1.5 + k = 3.07408523 Solving -1.5 + k = 3.07408523 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + k = 3.07408523 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + k = 3.07408523 + 1.5 k = 3.07408523 + 1.5 Combine like terms: 3.07408523 + 1.5 = 4.57408523 k = 4.57408523 Simplifying k = 4.57408523Subproblem 2
k + -1.5 = -3.07408523 Simplifying k + -1.5 = -3.07408523 Reorder the terms: -1.5 + k = -3.07408523 Solving -1.5 + k = -3.07408523 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + k = -3.07408523 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + k = -3.07408523 + 1.5 k = -3.07408523 + 1.5 Combine like terms: -3.07408523 + 1.5 = -1.57408523 k = -1.57408523 Simplifying k = -1.57408523Solution
The solution to the problem is based on the solutions from the subproblems. k = {4.57408523, -1.57408523}
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