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Simplifying 6k(k + 4) + 3 = 5(k2 + -12) + 8 Reorder the terms: 6k(4 + k) + 3 = 5(k2 + -12) + 8 (4 * 6k + k * 6k) + 3 = 5(k2 + -12) + 8 (24k + 6k2) + 3 = 5(k2 + -12) + 8 Reorder the terms: 3 + 24k + 6k2 = 5(k2 + -12) + 8 Reorder the terms: 3 + 24k + 6k2 = 5(-12 + k2) + 8 3 + 24k + 6k2 = (-12 * 5 + k2 * 5) + 8 3 + 24k + 6k2 = (-60 + 5k2) + 8 Reorder the terms: 3 + 24k + 6k2 = -60 + 8 + 5k2 Combine like terms: -60 + 8 = -52 3 + 24k + 6k2 = -52 + 5k2 Solving 3 + 24k + 6k2 = -52 + 5k2 Solving for variable 'k'. Reorder the terms: 3 + 52 + 24k + 6k2 + -5k2 = -52 + 5k2 + 52 + -5k2 Combine like terms: 3 + 52 = 55 55 + 24k + 6k2 + -5k2 = -52 + 5k2 + 52 + -5k2 Combine like terms: 6k2 + -5k2 = 1k2 55 + 24k + 1k2 = -52 + 5k2 + 52 + -5k2 Reorder the terms: 55 + 24k + 1k2 = -52 + 52 + 5k2 + -5k2 Combine like terms: -52 + 52 = 0 55 + 24k + 1k2 = 0 + 5k2 + -5k2 55 + 24k + 1k2 = 5k2 + -5k2 Combine like terms: 5k2 + -5k2 = 0 55 + 24k + 1k2 = 0 Begin completing the square. Move the constant term to the right: Add '-55' to each side of the equation. 55 + 24k + -55 + k2 = 0 + -55 Reorder the terms: 55 + -55 + 24k + k2 = 0 + -55 Combine like terms: 55 + -55 = 0 0 + 24k + k2 = 0 + -55 24k + k2 = 0 + -55 Combine like terms: 0 + -55 = -55 24k + k2 = -55 The k term is 24k. Take half its coefficient (12). Square it (144) and add it to both sides. Add '144' to each side of the equation. 24k + 144 + k2 = -55 + 144 Reorder the terms: 144 + 24k + k2 = -55 + 144 Combine like terms: -55 + 144 = 89 144 + 24k + k2 = 89 Factor a perfect square on the left side: (k + 12)(k + 12) = 89 Calculate the square root of the right side: 9.433981132 Break this problem into two subproblems by setting (k + 12) equal to 9.433981132 and -9.433981132.Subproblem 1
k + 12 = 9.433981132 Simplifying k + 12 = 9.433981132 Reorder the terms: 12 + k = 9.433981132 Solving 12 + k = 9.433981132 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + k = 9.433981132 + -12 Combine like terms: 12 + -12 = 0 0 + k = 9.433981132 + -12 k = 9.433981132 + -12 Combine like terms: 9.433981132 + -12 = -2.566018868 k = -2.566018868 Simplifying k = -2.566018868Subproblem 2
k + 12 = -9.433981132 Simplifying k + 12 = -9.433981132 Reorder the terms: 12 + k = -9.433981132 Solving 12 + k = -9.433981132 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + k = -9.433981132 + -12 Combine like terms: 12 + -12 = 0 0 + k = -9.433981132 + -12 k = -9.433981132 + -12 Combine like terms: -9.433981132 + -12 = -21.433981132 k = -21.433981132 Simplifying k = -21.433981132Solution
The solution to the problem is based on the solutions from the subproblems. k = {-2.566018868, -21.433981132}
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