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Simplifying 10(k + -2) + 7k(k + -3) = 0 Reorder the terms: 10(-2 + k) + 7k(k + -3) = 0 (-2 * 10 + k * 10) + 7k(k + -3) = 0 (-20 + 10k) + 7k(k + -3) = 0 Reorder the terms: -20 + 10k + 7k(-3 + k) = 0 -20 + 10k + (-3 * 7k + k * 7k) = 0 -20 + 10k + (-21k + 7k2) = 0 Combine like terms: 10k + -21k = -11k -20 + -11k + 7k2 = 0 Solving -20 + -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'. -2.857142857 + -1.571428571k + k2 = 0 Move the constant term to the right: Add '2.857142857' to each side of the equation. -2.857142857 + -1.571428571k + 2.857142857 + k2 = 0 + 2.857142857 Reorder the terms: -2.857142857 + 2.857142857 + -1.571428571k + k2 = 0 + 2.857142857 Combine like terms: -2.857142857 + 2.857142857 = 0.000000000 0.000000000 + -1.571428571k + k2 = 0 + 2.857142857 -1.571428571k + k2 = 0 + 2.857142857 Combine like terms: 0 + 2.857142857 = 2.857142857 -1.571428571k + k2 = 2.857142857 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 = 2.857142857 + 0.6173469384 Reorder the terms: 0.6173469384 + -1.571428571k + k2 = 2.857142857 + 0.6173469384 Combine like terms: 2.857142857 + 0.6173469384 = 3.4744897954 0.6173469384 + -1.571428571k + k2 = 3.4744897954 Factor a perfect square on the left side: (k + -0.7857142855)(k + -0.7857142855) = 3.4744897954 Calculate the square root of the right side: 1.863998336 Break this problem into two subproblems by setting (k + -0.7857142855) equal to 1.863998336 and -1.863998336.Subproblem 1
k + -0.7857142855 = 1.863998336 Simplifying k + -0.7857142855 = 1.863998336 Reorder the terms: -0.7857142855 + k = 1.863998336 Solving -0.7857142855 + k = 1.863998336 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 = 1.863998336 + 0.7857142855 Combine like terms: -0.7857142855 + 0.7857142855 = 0.0000000000 0.0000000000 + k = 1.863998336 + 0.7857142855 k = 1.863998336 + 0.7857142855 Combine like terms: 1.863998336 + 0.7857142855 = 2.6497126215 k = 2.6497126215 Simplifying k = 2.6497126215Subproblem 2
k + -0.7857142855 = -1.863998336 Simplifying k + -0.7857142855 = -1.863998336 Reorder the terms: -0.7857142855 + k = -1.863998336 Solving -0.7857142855 + k = -1.863998336 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 = -1.863998336 + 0.7857142855 Combine like terms: -0.7857142855 + 0.7857142855 = 0.0000000000 0.0000000000 + k = -1.863998336 + 0.7857142855 k = -1.863998336 + 0.7857142855 Combine like terms: -1.863998336 + 0.7857142855 = -1.0782840505 k = -1.0782840505 Simplifying k = -1.0782840505Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.6497126215, -1.0782840505}
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