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Simplifying 35k2 + -22k = 4 Reorder the terms: -22k + 35k2 = 4 Solving -22k + 35k2 = 4 Solving for variable 'k'. Reorder the terms: -4 + -22k + 35k2 = 4 + -4 Combine like terms: 4 + -4 = 0 -4 + -22k + 35k2 = 0 Begin completing the square. Divide all terms by 35 the coefficient of the squared term: Divide each side by '35'. -0.1142857143 + -0.6285714286k + k2 = 0 Move the constant term to the right: Add '0.1142857143' to each side of the equation. -0.1142857143 + -0.6285714286k + 0.1142857143 + k2 = 0 + 0.1142857143 Reorder the terms: -0.1142857143 + 0.1142857143 + -0.6285714286k + k2 = 0 + 0.1142857143 Combine like terms: -0.1142857143 + 0.1142857143 = 0.0000000000 0.0000000000 + -0.6285714286k + k2 = 0 + 0.1142857143 -0.6285714286k + k2 = 0 + 0.1142857143 Combine like terms: 0 + 0.1142857143 = 0.1142857143 -0.6285714286k + k2 = 0.1142857143 The k term is -0.6285714286k. Take half its coefficient (-0.3142857143). Square it (0.09877551021) and add it to both sides. Add '0.09877551021' to each side of the equation. -0.6285714286k + 0.09877551021 + k2 = 0.1142857143 + 0.09877551021 Reorder the terms: 0.09877551021 + -0.6285714286k + k2 = 0.1142857143 + 0.09877551021 Combine like terms: 0.1142857143 + 0.09877551021 = 0.21306122451 0.09877551021 + -0.6285714286k + k2 = 0.21306122451 Factor a perfect square on the left side: (k + -0.3142857143)(k + -0.3142857143) = 0.21306122451 Calculate the square root of the right side: 0.461585555 Break this problem into two subproblems by setting (k + -0.3142857143) equal to 0.461585555 and -0.461585555.Subproblem 1
k + -0.3142857143 = 0.461585555 Simplifying k + -0.3142857143 = 0.461585555 Reorder the terms: -0.3142857143 + k = 0.461585555 Solving -0.3142857143 + k = 0.461585555 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.3142857143' to each side of the equation. -0.3142857143 + 0.3142857143 + k = 0.461585555 + 0.3142857143 Combine like terms: -0.3142857143 + 0.3142857143 = 0.0000000000 0.0000000000 + k = 0.461585555 + 0.3142857143 k = 0.461585555 + 0.3142857143 Combine like terms: 0.461585555 + 0.3142857143 = 0.7758712693 k = 0.7758712693 Simplifying k = 0.7758712693Subproblem 2
k + -0.3142857143 = -0.461585555 Simplifying k + -0.3142857143 = -0.461585555 Reorder the terms: -0.3142857143 + k = -0.461585555 Solving -0.3142857143 + k = -0.461585555 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.3142857143' to each side of the equation. -0.3142857143 + 0.3142857143 + k = -0.461585555 + 0.3142857143 Combine like terms: -0.3142857143 + 0.3142857143 = 0.0000000000 0.0000000000 + k = -0.461585555 + 0.3142857143 k = -0.461585555 + 0.3142857143 Combine like terms: -0.461585555 + 0.3142857143 = -0.1472998407 k = -0.1472998407 Simplifying k = -0.1472998407Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.7758712693, -0.1472998407}
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