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Simplifying 21k2 + -37k = 12 Reorder the terms: -37k + 21k2 = 12 Solving -37k + 21k2 = 12 Solving for variable 'k'. Reorder the terms: -12 + -37k + 21k2 = 12 + -12 Combine like terms: 12 + -12 = 0 -12 + -37k + 21k2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -0.5714285714 + -1.761904762k + k2 = 0 Move the constant term to the right: Add '0.5714285714' to each side of the equation. -0.5714285714 + -1.761904762k + 0.5714285714 + k2 = 0 + 0.5714285714 Reorder the terms: -0.5714285714 + 0.5714285714 + -1.761904762k + k2 = 0 + 0.5714285714 Combine like terms: -0.5714285714 + 0.5714285714 = 0.0000000000 0.0000000000 + -1.761904762k + k2 = 0 + 0.5714285714 -1.761904762k + k2 = 0 + 0.5714285714 Combine like terms: 0 + 0.5714285714 = 0.5714285714 -1.761904762k + k2 = 0.5714285714 The k term is -1.761904762k. Take half its coefficient (-0.880952381). Square it (0.7760770976) and add it to both sides. Add '0.7760770976' to each side of the equation. -1.761904762k + 0.7760770976 + k2 = 0.5714285714 + 0.7760770976 Reorder the terms: 0.7760770976 + -1.761904762k + k2 = 0.5714285714 + 0.7760770976 Combine like terms: 0.5714285714 + 0.7760770976 = 1.347505669 0.7760770976 + -1.761904762k + k2 = 1.347505669 Factor a perfect square on the left side: (k + -0.880952381)(k + -0.880952381) = 1.347505669 Calculate the square root of the right side: 1.160821118 Break this problem into two subproblems by setting (k + -0.880952381) equal to 1.160821118 and -1.160821118.Subproblem 1
k + -0.880952381 = 1.160821118 Simplifying k + -0.880952381 = 1.160821118 Reorder the terms: -0.880952381 + k = 1.160821118 Solving -0.880952381 + k = 1.160821118 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.880952381' to each side of the equation. -0.880952381 + 0.880952381 + k = 1.160821118 + 0.880952381 Combine like terms: -0.880952381 + 0.880952381 = 0.000000000 0.000000000 + k = 1.160821118 + 0.880952381 k = 1.160821118 + 0.880952381 Combine like terms: 1.160821118 + 0.880952381 = 2.041773499 k = 2.041773499 Simplifying k = 2.041773499Subproblem 2
k + -0.880952381 = -1.160821118 Simplifying k + -0.880952381 = -1.160821118 Reorder the terms: -0.880952381 + k = -1.160821118 Solving -0.880952381 + k = -1.160821118 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.880952381' to each side of the equation. -0.880952381 + 0.880952381 + k = -1.160821118 + 0.880952381 Combine like terms: -0.880952381 + 0.880952381 = 0.000000000 0.000000000 + k = -1.160821118 + 0.880952381 k = -1.160821118 + 0.880952381 Combine like terms: -1.160821118 + 0.880952381 = -0.279868737 k = -0.279868737 Simplifying k = -0.279868737Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.041773499, -0.279868737}
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