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Simplifying d2 + -8d + 16 = 11 Reorder the terms: 16 + -8d + d2 = 11 Solving 16 + -8d + d2 = 11 Solving for variable 'd'. Reorder the terms: 16 + -11 + -8d + d2 = 11 + -11 Combine like terms: 16 + -11 = 5 5 + -8d + d2 = 11 + -11 Combine like terms: 11 + -11 = 0 5 + -8d + d2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + -8d + -5 + d2 = 0 + -5 Reorder the terms: 5 + -5 + -8d + d2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + -8d + d2 = 0 + -5 -8d + d2 = 0 + -5 Combine like terms: 0 + -5 = -5 -8d + d2 = -5 The d term is -8d. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8d + 16 + d2 = -5 + 16 Reorder the terms: 16 + -8d + d2 = -5 + 16 Combine like terms: -5 + 16 = 11 16 + -8d + d2 = 11 Factor a perfect square on the left side: (d + -4)(d + -4) = 11 Calculate the square root of the right side: 3.31662479 Break this problem into two subproblems by setting (d + -4) equal to 3.31662479 and -3.31662479.Subproblem 1
d + -4 = 3.31662479 Simplifying d + -4 = 3.31662479 Reorder the terms: -4 + d = 3.31662479 Solving -4 + d = 3.31662479 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + d = 3.31662479 + 4 Combine like terms: -4 + 4 = 0 0 + d = 3.31662479 + 4 d = 3.31662479 + 4 Combine like terms: 3.31662479 + 4 = 7.31662479 d = 7.31662479 Simplifying d = 7.31662479Subproblem 2
d + -4 = -3.31662479 Simplifying d + -4 = -3.31662479 Reorder the terms: -4 + d = -3.31662479 Solving -4 + d = -3.31662479 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + d = -3.31662479 + 4 Combine like terms: -4 + 4 = 0 0 + d = -3.31662479 + 4 d = -3.31662479 + 4 Combine like terms: -3.31662479 + 4 = 0.68337521 d = 0.68337521 Simplifying d = 0.68337521Solution
The solution to the problem is based on the solutions from the subproblems. d = {7.31662479, 0.68337521}
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