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Simplifying k2 + -8k + -84 = 9 Reorder the terms: -84 + -8k + k2 = 9 Solving -84 + -8k + k2 = 9 Solving for variable 'k'. Reorder the terms: -84 + -9 + -8k + k2 = 9 + -9 Combine like terms: -84 + -9 = -93 -93 + -8k + k2 = 9 + -9 Combine like terms: 9 + -9 = 0 -93 + -8k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '93' to each side of the equation. -93 + -8k + 93 + k2 = 0 + 93 Reorder the terms: -93 + 93 + -8k + k2 = 0 + 93 Combine like terms: -93 + 93 = 0 0 + -8k + k2 = 0 + 93 -8k + k2 = 0 + 93 Combine like terms: 0 + 93 = 93 -8k + k2 = 93 The k term is -8k. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8k + 16 + k2 = 93 + 16 Reorder the terms: 16 + -8k + k2 = 93 + 16 Combine like terms: 93 + 16 = 109 16 + -8k + k2 = 109 Factor a perfect square on the left side: (k + -4)(k + -4) = 109 Calculate the square root of the right side: 10.440306509 Break this problem into two subproblems by setting (k + -4) equal to 10.440306509 and -10.440306509.Subproblem 1
k + -4 = 10.440306509 Simplifying k + -4 = 10.440306509 Reorder the terms: -4 + k = 10.440306509 Solving -4 + k = 10.440306509 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + k = 10.440306509 + 4 Combine like terms: -4 + 4 = 0 0 + k = 10.440306509 + 4 k = 10.440306509 + 4 Combine like terms: 10.440306509 + 4 = 14.440306509 k = 14.440306509 Simplifying k = 14.440306509Subproblem 2
k + -4 = -10.440306509 Simplifying k + -4 = -10.440306509 Reorder the terms: -4 + k = -10.440306509 Solving -4 + k = -10.440306509 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + k = -10.440306509 + 4 Combine like terms: -4 + 4 = 0 0 + k = -10.440306509 + 4 k = -10.440306509 + 4 Combine like terms: -10.440306509 + 4 = -6.440306509 k = -6.440306509 Simplifying k = -6.440306509Solution
The solution to the problem is based on the solutions from the subproblems. k = {14.440306509, -6.440306509}
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