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Simplifying (5k2 + -7k + -8) + (6k2 + 9k + -10) = 0 Reorder the terms: (-8 + -7k + 5k2) + (6k2 + 9k + -10) = 0 Remove parenthesis around (-8 + -7k + 5k2) -8 + -7k + 5k2 + (6k2 + 9k + -10) = 0 Reorder the terms: -8 + -7k + 5k2 + (-10 + 9k + 6k2) = 0 Remove parenthesis around (-10 + 9k + 6k2) -8 + -7k + 5k2 + -10 + 9k + 6k2 = 0 Reorder the terms: -8 + -10 + -7k + 9k + 5k2 + 6k2 = 0 Combine like terms: -8 + -10 = -18 -18 + -7k + 9k + 5k2 + 6k2 = 0 Combine like terms: -7k + 9k = 2k -18 + 2k + 5k2 + 6k2 = 0 Combine like terms: 5k2 + 6k2 = 11k2 -18 + 2k + 11k2 = 0 Solving -18 + 2k + 11k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -1.636363636 + 0.1818181818k + k2 = 0 Move the constant term to the right: Add '1.636363636' to each side of the equation. -1.636363636 + 0.1818181818k + 1.636363636 + k2 = 0 + 1.636363636 Reorder the terms: -1.636363636 + 1.636363636 + 0.1818181818k + k2 = 0 + 1.636363636 Combine like terms: -1.636363636 + 1.636363636 = 0.000000000 0.000000000 + 0.1818181818k + k2 = 0 + 1.636363636 0.1818181818k + k2 = 0 + 1.636363636 Combine like terms: 0 + 1.636363636 = 1.636363636 0.1818181818k + k2 = 1.636363636 The k term is 0.1818181818k. Take half its coefficient (0.0909090909). Square it (0.008264462808) and add it to both sides. Add '0.008264462808' to each side of the equation. 0.1818181818k + 0.008264462808 + k2 = 1.636363636 + 0.008264462808 Reorder the terms: 0.008264462808 + 0.1818181818k + k2 = 1.636363636 + 0.008264462808 Combine like terms: 1.636363636 + 0.008264462808 = 1.644628098808 0.008264462808 + 0.1818181818k + k2 = 1.644628098808 Factor a perfect square on the left side: (k + 0.0909090909)(k + 0.0909090909) = 1.644628098808 Calculate the square root of the right side: 1.282430543 Break this problem into two subproblems by setting (k + 0.0909090909) equal to 1.282430543 and -1.282430543.Subproblem 1
k + 0.0909090909 = 1.282430543 Simplifying k + 0.0909090909 = 1.282430543 Reorder the terms: 0.0909090909 + k = 1.282430543 Solving 0.0909090909 + k = 1.282430543 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.0909090909' to each side of the equation. 0.0909090909 + -0.0909090909 + k = 1.282430543 + -0.0909090909 Combine like terms: 0.0909090909 + -0.0909090909 = 0.0000000000 0.0000000000 + k = 1.282430543 + -0.0909090909 k = 1.282430543 + -0.0909090909 Combine like terms: 1.282430543 + -0.0909090909 = 1.1915214521 k = 1.1915214521 Simplifying k = 1.1915214521Subproblem 2
k + 0.0909090909 = -1.282430543 Simplifying k + 0.0909090909 = -1.282430543 Reorder the terms: 0.0909090909 + k = -1.282430543 Solving 0.0909090909 + k = -1.282430543 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.0909090909' to each side of the equation. 0.0909090909 + -0.0909090909 + k = -1.282430543 + -0.0909090909 Combine like terms: 0.0909090909 + -0.0909090909 = 0.0000000000 0.0000000000 + k = -1.282430543 + -0.0909090909 k = -1.282430543 + -0.0909090909 Combine like terms: -1.282430543 + -0.0909090909 = -1.3733396339 k = -1.3733396339 Simplifying k = -1.3733396339Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.1915214521, -1.3733396339}
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