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Simplifying d2 + 24d + 144 = 8 Reorder the terms: 144 + 24d + d2 = 8 Solving 144 + 24d + d2 = 8 Solving for variable 'd'. Reorder the terms: 144 + -8 + 24d + d2 = 8 + -8 Combine like terms: 144 + -8 = 136 136 + 24d + d2 = 8 + -8 Combine like terms: 8 + -8 = 0 136 + 24d + d2 = 0 Begin completing the square. Move the constant term to the right: Add '-136' to each side of the equation. 136 + 24d + -136 + d2 = 0 + -136 Reorder the terms: 136 + -136 + 24d + d2 = 0 + -136 Combine like terms: 136 + -136 = 0 0 + 24d + d2 = 0 + -136 24d + d2 = 0 + -136 Combine like terms: 0 + -136 = -136 24d + d2 = -136 The d term is 24d. Take half its coefficient (12). Square it (144) and add it to both sides. Add '144' to each side of the equation. 24d + 144 + d2 = -136 + 144 Reorder the terms: 144 + 24d + d2 = -136 + 144 Combine like terms: -136 + 144 = 8 144 + 24d + d2 = 8 Factor a perfect square on the left side: (d + 12)(d + 12) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (d + 12) equal to 2.828427125 and -2.828427125.Subproblem 1
d + 12 = 2.828427125 Simplifying d + 12 = 2.828427125 Reorder the terms: 12 + d = 2.828427125 Solving 12 + d = 2.828427125 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + d = 2.828427125 + -12 Combine like terms: 12 + -12 = 0 0 + d = 2.828427125 + -12 d = 2.828427125 + -12 Combine like terms: 2.828427125 + -12 = -9.171572875 d = -9.171572875 Simplifying d = -9.171572875Subproblem 2
d + 12 = -2.828427125 Simplifying d + 12 = -2.828427125 Reorder the terms: 12 + d = -2.828427125 Solving 12 + d = -2.828427125 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + d = -2.828427125 + -12 Combine like terms: 12 + -12 = 0 0 + d = -2.828427125 + -12 d = -2.828427125 + -12 Combine like terms: -2.828427125 + -12 = -14.828427125 d = -14.828427125 Simplifying d = -14.828427125Solution
The solution to the problem is based on the solutions from the subproblems. d = {-9.171572875, -14.828427125}
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