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Simplifying (4k + -3)(4k + -3) = 50 Reorder the terms: (-3 + 4k)(4k + -3) = 50 Reorder the terms: (-3 + 4k)(-3 + 4k) = 50 Multiply (-3 + 4k) * (-3 + 4k) (-3(-3 + 4k) + 4k * (-3 + 4k)) = 50 ((-3 * -3 + 4k * -3) + 4k * (-3 + 4k)) = 50 ((9 + -12k) + 4k * (-3 + 4k)) = 50 (9 + -12k + (-3 * 4k + 4k * 4k)) = 50 (9 + -12k + (-12k + 16k2)) = 50 Combine like terms: -12k + -12k = -24k (9 + -24k + 16k2) = 50 Solving 9 + -24k + 16k2 = 50 Solving for variable 'k'. Reorder the terms: 9 + -50 + -24k + 16k2 = 50 + -50 Combine like terms: 9 + -50 = -41 -41 + -24k + 16k2 = 50 + -50 Combine like terms: 50 + -50 = 0 -41 + -24k + 16k2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -2.5625 + -1.5k + k2 = 0 Move the constant term to the right: Add '2.5625' to each side of the equation. -2.5625 + -1.5k + 2.5625 + k2 = 0 + 2.5625 Reorder the terms: -2.5625 + 2.5625 + -1.5k + k2 = 0 + 2.5625 Combine like terms: -2.5625 + 2.5625 = 0.0000 0.0000 + -1.5k + k2 = 0 + 2.5625 -1.5k + k2 = 0 + 2.5625 Combine like terms: 0 + 2.5625 = 2.5625 -1.5k + k2 = 2.5625 The k term is -1.5k. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5k + 0.5625 + k2 = 2.5625 + 0.5625 Reorder the terms: 0.5625 + -1.5k + k2 = 2.5625 + 0.5625 Combine like terms: 2.5625 + 0.5625 = 3.125 0.5625 + -1.5k + k2 = 3.125 Factor a perfect square on the left side: (k + -0.75)(k + -0.75) = 3.125 Calculate the square root of the right side: 1.767766953 Break this problem into two subproblems by setting (k + -0.75) equal to 1.767766953 and -1.767766953.Subproblem 1
k + -0.75 = 1.767766953 Simplifying k + -0.75 = 1.767766953 Reorder the terms: -0.75 + k = 1.767766953 Solving -0.75 + k = 1.767766953 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + k = 1.767766953 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + k = 1.767766953 + 0.75 k = 1.767766953 + 0.75 Combine like terms: 1.767766953 + 0.75 = 2.517766953 k = 2.517766953 Simplifying k = 2.517766953Subproblem 2
k + -0.75 = -1.767766953 Simplifying k + -0.75 = -1.767766953 Reorder the terms: -0.75 + k = -1.767766953 Solving -0.75 + k = -1.767766953 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + k = -1.767766953 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + k = -1.767766953 + 0.75 k = -1.767766953 + 0.75 Combine like terms: -1.767766953 + 0.75 = -1.017766953 k = -1.017766953 Simplifying k = -1.017766953Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.517766953, -1.017766953}
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