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Simplifying 3k2 + 72 = 32k Reorder the terms: 72 + 3k2 = 32k Solving 72 + 3k2 = 32k Solving for variable 'k'. Reorder the terms: 72 + -32k + 3k2 = 32k + -32k Combine like terms: 32k + -32k = 0 72 + -32k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 24 + -10.66666667k + k2 = 0 Move the constant term to the right: Add '-24' to each side of the equation. 24 + -10.66666667k + -24 + k2 = 0 + -24 Reorder the terms: 24 + -24 + -10.66666667k + k2 = 0 + -24 Combine like terms: 24 + -24 = 0 0 + -10.66666667k + k2 = 0 + -24 -10.66666667k + k2 = 0 + -24 Combine like terms: 0 + -24 = -24 -10.66666667k + k2 = -24 The k term is -10.66666667k. Take half its coefficient (-5.333333335). Square it (28.44444446) and add it to both sides. Add '28.44444446' to each side of the equation. -10.66666667k + 28.44444446 + k2 = -24 + 28.44444446 Reorder the terms: 28.44444446 + -10.66666667k + k2 = -24 + 28.44444446 Combine like terms: -24 + 28.44444446 = 4.44444446 28.44444446 + -10.66666667k + k2 = 4.44444446 Factor a perfect square on the left side: (k + -5.333333335)(k + -5.333333335) = 4.44444446 Calculate the square root of the right side: 2.10818511 Break this problem into two subproblems by setting (k + -5.333333335) equal to 2.10818511 and -2.10818511.Subproblem 1
k + -5.333333335 = 2.10818511 Simplifying k + -5.333333335 = 2.10818511 Reorder the terms: -5.333333335 + k = 2.10818511 Solving -5.333333335 + k = 2.10818511 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '5.333333335' to each side of the equation. -5.333333335 + 5.333333335 + k = 2.10818511 + 5.333333335 Combine like terms: -5.333333335 + 5.333333335 = 0.000000000 0.000000000 + k = 2.10818511 + 5.333333335 k = 2.10818511 + 5.333333335 Combine like terms: 2.10818511 + 5.333333335 = 7.441518445 k = 7.441518445 Simplifying k = 7.441518445Subproblem 2
k + -5.333333335 = -2.10818511 Simplifying k + -5.333333335 = -2.10818511 Reorder the terms: -5.333333335 + k = -2.10818511 Solving -5.333333335 + k = -2.10818511 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '5.333333335' to each side of the equation. -5.333333335 + 5.333333335 + k = -2.10818511 + 5.333333335 Combine like terms: -5.333333335 + 5.333333335 = 0.000000000 0.000000000 + k = -2.10818511 + 5.333333335 k = -2.10818511 + 5.333333335 Combine like terms: -2.10818511 + 5.333333335 = 3.225148225 k = 3.225148225 Simplifying k = 3.225148225Solution
The solution to the problem is based on the solutions from the subproblems. k = {7.441518445, 3.225148225}
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