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Simplifying (3k + -1)(k + 4) = 11 Reorder the terms: (-1 + 3k)(k + 4) = 11 Reorder the terms: (-1 + 3k)(4 + k) = 11 Multiply (-1 + 3k) * (4 + k) (-1(4 + k) + 3k * (4 + k)) = 11 ((4 * -1 + k * -1) + 3k * (4 + k)) = 11 ((-4 + -1k) + 3k * (4 + k)) = 11 (-4 + -1k + (4 * 3k + k * 3k)) = 11 (-4 + -1k + (12k + 3k2)) = 11 Combine like terms: -1k + 12k = 11k (-4 + 11k + 3k2) = 11 Solving -4 + 11k + 3k2 = 11 Solving for variable 'k'. Reorder the terms: -4 + -11 + 11k + 3k2 = 11 + -11 Combine like terms: -4 + -11 = -15 -15 + 11k + 3k2 = 11 + -11 Combine like terms: 11 + -11 = 0 -15 + 11k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -5 + 3.666666667k + k2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 3.666666667k + 5 + k2 = 0 + 5 Reorder the terms: -5 + 5 + 3.666666667k + k2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 3.666666667k + k2 = 0 + 5 3.666666667k + k2 = 0 + 5 Combine like terms: 0 + 5 = 5 3.666666667k + k2 = 5 The k term is 3.666666667k. Take half its coefficient (1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. 3.666666667k + 3.361111114 + k2 = 5 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667k + k2 = 5 + 3.361111114 Combine like terms: 5 + 3.361111114 = 8.361111114 3.361111114 + 3.666666667k + k2 = 8.361111114 Factor a perfect square on the left side: (k + 1.833333334)(k + 1.833333334) = 8.361111114 Calculate the square root of the right side: 2.891558596 Break this problem into two subproblems by setting (k + 1.833333334) equal to 2.891558596 and -2.891558596.Subproblem 1
k + 1.833333334 = 2.891558596 Simplifying k + 1.833333334 = 2.891558596 Reorder the terms: 1.833333334 + k = 2.891558596 Solving 1.833333334 + k = 2.891558596 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + k = 2.891558596 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + k = 2.891558596 + -1.833333334 k = 2.891558596 + -1.833333334 Combine like terms: 2.891558596 + -1.833333334 = 1.058225262 k = 1.058225262 Simplifying k = 1.058225262Subproblem 2
k + 1.833333334 = -2.891558596 Simplifying k + 1.833333334 = -2.891558596 Reorder the terms: 1.833333334 + k = -2.891558596 Solving 1.833333334 + k = -2.891558596 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + k = -2.891558596 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + k = -2.891558596 + -1.833333334 k = -2.891558596 + -1.833333334 Combine like terms: -2.891558596 + -1.833333334 = -4.72489193 k = -4.72489193 Simplifying k = -4.72489193Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.058225262, -4.72489193}
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