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Simplifying 4k2 + 4k + 1 = 19(4k2 + 4k + 1) Reorder the terms: 1 + 4k + 4k2 = 19(4k2 + 4k + 1) Reorder the terms: 1 + 4k + 4k2 = 19(1 + 4k + 4k2) 1 + 4k + 4k2 = (1 * 19 + 4k * 19 + 4k2 * 19) 1 + 4k + 4k2 = (19 + 76k + 76k2) Solving 1 + 4k + 4k2 = 19 + 76k + 76k2 Solving for variable 'k'. Reorder the terms: 1 + -19 + 4k + -76k + 4k2 + -76k2 = 19 + 76k + 76k2 + -19 + -76k + -76k2 Combine like terms: 1 + -19 = -18 -18 + 4k + -76k + 4k2 + -76k2 = 19 + 76k + 76k2 + -19 + -76k + -76k2 Combine like terms: 4k + -76k = -72k -18 + -72k + 4k2 + -76k2 = 19 + 76k + 76k2 + -19 + -76k + -76k2 Combine like terms: 4k2 + -76k2 = -72k2 -18 + -72k + -72k2 = 19 + 76k + 76k2 + -19 + -76k + -76k2 Reorder the terms: -18 + -72k + -72k2 = 19 + -19 + 76k + -76k + 76k2 + -76k2 Combine like terms: 19 + -19 = 0 -18 + -72k + -72k2 = 0 + 76k + -76k + 76k2 + -76k2 -18 + -72k + -72k2 = 76k + -76k + 76k2 + -76k2 Combine like terms: 76k + -76k = 0 -18 + -72k + -72k2 = 0 + 76k2 + -76k2 -18 + -72k + -72k2 = 76k2 + -76k2 Combine like terms: 76k2 + -76k2 = 0 -18 + -72k + -72k2 = 0 Factor out the Greatest Common Factor (GCF), '-18'. -18(1 + 4k + 4k2) = 0 Factor a trinomial. -18((1 + 2k)(1 + 2k)) = 0 Ignore the factor -18.Subproblem 1
Set the factor '(1 + 2k)' equal to zero and attempt to solve: Simplifying 1 + 2k = 0 Solving 1 + 2k = 0 Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + 2k = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 2k = 0 + -1 2k = 0 + -1 Combine like terms: 0 + -1 = -1 2k = -1 Divide each side by '2'. k = -0.5 Simplifying k = -0.5Subproblem 2
Set the factor '(1 + 2k)' equal to zero and attempt to solve: Simplifying 1 + 2k = 0 Solving 1 + 2k = 0 Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + 2k = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 2k = 0 + -1 2k = 0 + -1 Combine like terms: 0 + -1 = -1 2k = -1 Divide each side by '2'. k = -0.5 Simplifying k = -0.5Solution
k = {-0.5, -0.5}
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