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Simplifying 1.33k2 + -16k + 28 = 0 Reorder the terms: 28 + -16k + 1.33k2 = 0 Solving 28 + -16k + 1.33k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 1.33 the coefficient of the squared term: Divide each side by '1.33'. 21.05263158 + -12.03007519k + k2 = 0 Move the constant term to the right: Add '-21.05263158' to each side of the equation. 21.05263158 + -12.03007519k + -21.05263158 + k2 = 0 + -21.05263158 Reorder the terms: 21.05263158 + -21.05263158 + -12.03007519k + k2 = 0 + -21.05263158 Combine like terms: 21.05263158 + -21.05263158 = 0.00000000 0.00000000 + -12.03007519k + k2 = 0 + -21.05263158 -12.03007519k + k2 = 0 + -21.05263158 Combine like terms: 0 + -21.05263158 = -21.05263158 -12.03007519k + k2 = -21.05263158 The k term is -12.03007519k. Take half its coefficient (-6.015037595). Square it (36.18067727) and add it to both sides. Add '36.18067727' to each side of the equation. -12.03007519k + 36.18067727 + k2 = -21.05263158 + 36.18067727 Reorder the terms: 36.18067727 + -12.03007519k + k2 = -21.05263158 + 36.18067727 Combine like terms: -21.05263158 + 36.18067727 = 15.12804569 36.18067727 + -12.03007519k + k2 = 15.12804569 Factor a perfect square on the left side: (k + -6.015037595)(k + -6.015037595) = 15.12804569 Calculate the square root of the right side: 3.889478846 Break this problem into two subproblems by setting (k + -6.015037595) equal to 3.889478846 and -3.889478846.Subproblem 1
k + -6.015037595 = 3.889478846 Simplifying k + -6.015037595 = 3.889478846 Reorder the terms: -6.015037595 + k = 3.889478846 Solving -6.015037595 + k = 3.889478846 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '6.015037595' to each side of the equation. -6.015037595 + 6.015037595 + k = 3.889478846 + 6.015037595 Combine like terms: -6.015037595 + 6.015037595 = 0.000000000 0.000000000 + k = 3.889478846 + 6.015037595 k = 3.889478846 + 6.015037595 Combine like terms: 3.889478846 + 6.015037595 = 9.904516441 k = 9.904516441 Simplifying k = 9.904516441Subproblem 2
k + -6.015037595 = -3.889478846 Simplifying k + -6.015037595 = -3.889478846 Reorder the terms: -6.015037595 + k = -3.889478846 Solving -6.015037595 + k = -3.889478846 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '6.015037595' to each side of the equation. -6.015037595 + 6.015037595 + k = -3.889478846 + 6.015037595 Combine like terms: -6.015037595 + 6.015037595 = 0.000000000 0.000000000 + k = -3.889478846 + 6.015037595 k = -3.889478846 + 6.015037595 Combine like terms: -3.889478846 + 6.015037595 = 2.125558749 k = 2.125558749 Simplifying k = 2.125558749Solution
The solution to the problem is based on the solutions from the subproblems. k = {9.904516441, 2.125558749}
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