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Simplifying 7k2 + -11k + -3 = 0 Reorder the terms: -3 + -11k + 7k2 = 0 Solving -3 + -11k + 7k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.4285714286 + -1.571428571k + k2 = 0 Move the constant term to the right: Add '0.4285714286' to each side of the equation. -0.4285714286 + -1.571428571k + 0.4285714286 + k2 = 0 + 0.4285714286 Reorder the terms: -0.4285714286 + 0.4285714286 + -1.571428571k + k2 = 0 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + -1.571428571k + k2 = 0 + 0.4285714286 -1.571428571k + k2 = 0 + 0.4285714286 Combine like terms: 0 + 0.4285714286 = 0.4285714286 -1.571428571k + k2 = 0.4285714286 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 = 0.4285714286 + 0.6173469384 Reorder the terms: 0.6173469384 + -1.571428571k + k2 = 0.4285714286 + 0.6173469384 Combine like terms: 0.4285714286 + 0.6173469384 = 1.045918367 0.6173469384 + -1.571428571k + k2 = 1.045918367 Factor a perfect square on the left side: (k + -0.7857142855)(k + -0.7857142855) = 1.045918367 Calculate the square root of the right side: 1.022701504 Break this problem into two subproblems by setting (k + -0.7857142855) equal to 1.022701504 and -1.022701504.Subproblem 1
k + -0.7857142855 = 1.022701504 Simplifying k + -0.7857142855 = 1.022701504 Reorder the terms: -0.7857142855 + k = 1.022701504 Solving -0.7857142855 + k = 1.022701504 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 = 1.022701504 + 0.7857142855 Combine like terms: -0.7857142855 + 0.7857142855 = 0.0000000000 0.0000000000 + k = 1.022701504 + 0.7857142855 k = 1.022701504 + 0.7857142855 Combine like terms: 1.022701504 + 0.7857142855 = 1.8084157895 k = 1.8084157895 Simplifying k = 1.8084157895Subproblem 2
k + -0.7857142855 = -1.022701504 Simplifying k + -0.7857142855 = -1.022701504 Reorder the terms: -0.7857142855 + k = -1.022701504 Solving -0.7857142855 + k = -1.022701504 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 = -1.022701504 + 0.7857142855 Combine like terms: -0.7857142855 + 0.7857142855 = 0.0000000000 0.0000000000 + k = -1.022701504 + 0.7857142855 k = -1.022701504 + 0.7857142855 Combine like terms: -1.022701504 + 0.7857142855 = -0.2369872185 k = -0.2369872185 Simplifying k = -0.2369872185Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.8084157895, -0.2369872185}
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