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Simplifying 7k2 + -40 = -11k Reorder the terms: -40 + 7k2 = -11k Solving -40 + 7k2 = -11k Solving for variable 'k'. Reorder the terms: -40 + 11k + 7k2 = -11k + 11k Combine like terms: -11k + 11k = 0 -40 + 11k + 7k2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -5.714285714 + 1.571428571k + k2 = 0 Move the constant term to the right: Add '5.714285714' to each side of the equation. -5.714285714 + 1.571428571k + 5.714285714 + k2 = 0 + 5.714285714 Reorder the terms: -5.714285714 + 5.714285714 + 1.571428571k + k2 = 0 + 5.714285714 Combine like terms: -5.714285714 + 5.714285714 = 0.000000000 0.000000000 + 1.571428571k + k2 = 0 + 5.714285714 1.571428571k + k2 = 0 + 5.714285714 Combine like terms: 0 + 5.714285714 = 5.714285714 1.571428571k + k2 = 5.714285714 The k term is 1.571428571k. Take half its coefficient (0.7857142855). Square it (0.6173469384) and add it to both sides. Add '0.6173469384' to each side of the equation. 1.571428571k + 0.6173469384 + k2 = 5.714285714 + 0.6173469384 Reorder the terms: 0.6173469384 + 1.571428571k + k2 = 5.714285714 + 0.6173469384 Combine like terms: 5.714285714 + 0.6173469384 = 6.3316326524 0.6173469384 + 1.571428571k + k2 = 6.3316326524 Factor a perfect square on the left side: (k + 0.7857142855)(k + 0.7857142855) = 6.3316326524 Calculate the square root of the right side: 2.516273565 Break this problem into two subproblems by setting (k + 0.7857142855) equal to 2.516273565 and -2.516273565.Subproblem 1
k + 0.7857142855 = 2.516273565 Simplifying k + 0.7857142855 = 2.516273565 Reorder the terms: 0.7857142855 + k = 2.516273565 Solving 0.7857142855 + k = 2.516273565 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.7857142855' to each side of the equation. 0.7857142855 + -0.7857142855 + k = 2.516273565 + -0.7857142855 Combine like terms: 0.7857142855 + -0.7857142855 = 0.0000000000 0.0000000000 + k = 2.516273565 + -0.7857142855 k = 2.516273565 + -0.7857142855 Combine like terms: 2.516273565 + -0.7857142855 = 1.7305592795 k = 1.7305592795 Simplifying k = 1.7305592795Subproblem 2
k + 0.7857142855 = -2.516273565 Simplifying k + 0.7857142855 = -2.516273565 Reorder the terms: 0.7857142855 + k = -2.516273565 Solving 0.7857142855 + k = -2.516273565 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.7857142855' to each side of the equation. 0.7857142855 + -0.7857142855 + k = -2.516273565 + -0.7857142855 Combine like terms: 0.7857142855 + -0.7857142855 = 0.0000000000 0.0000000000 + k = -2.516273565 + -0.7857142855 k = -2.516273565 + -0.7857142855 Combine like terms: -2.516273565 + -0.7857142855 = -3.3019878505 k = -3.3019878505 Simplifying k = -3.3019878505Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.7305592795, -3.3019878505}
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