18d^2-89d=-84

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Solution for 18d^2-89d=-84 equation:


Simplifying
18d2 + -89d = -84

Reorder the terms:
-89d + 18d2 = -84

Solving
-89d + 18d2 = -84

Solving for variable 'd'.

Reorder the terms:
84 + -89d + 18d2 = -84 + 84

Combine like terms: -84 + 84 = 0
84 + -89d + 18d2 = 0

Begin completing the square.  Divide all terms by
18 the coefficient of the squared term: 

Divide each side by '18'.
4.666666667 + -4.944444444d + d2 = 0

Move the constant term to the right:

Add '-4.666666667' to each side of the equation.
4.666666667 + -4.944444444d + -4.666666667 + d2 = 0 + -4.666666667

Reorder the terms:
4.666666667 + -4.666666667 + -4.944444444d + d2 = 0 + -4.666666667

Combine like terms: 4.666666667 + -4.666666667 = 0.000000000
0.000000000 + -4.944444444d + d2 = 0 + -4.666666667
-4.944444444d + d2 = 0 + -4.666666667

Combine like terms: 0 + -4.666666667 = -4.666666667
-4.944444444d + d2 = -4.666666667

The d term is -4.944444444d.  Take half its coefficient (-2.472222222).
Square it (6.111882715) and add it to both sides.

Add '6.111882715' to each side of the equation.
-4.944444444d + 6.111882715 + d2 = -4.666666667 + 6.111882715

Reorder the terms:
6.111882715 + -4.944444444d + d2 = -4.666666667 + 6.111882715

Combine like terms: -4.666666667 + 6.111882715 = 1.445216048
6.111882715 + -4.944444444d + d2 = 1.445216048

Factor a perfect square on the left side:
(d + -2.472222222)(d + -2.472222222) = 1.445216048

Calculate the square root of the right side: 1.202171389

Break this problem into two subproblems by setting 
(d + -2.472222222) equal to 1.202171389 and -1.202171389.

Subproblem 1

d + -2.472222222 = 1.202171389 Simplifying d + -2.472222222 = 1.202171389 Reorder the terms: -2.472222222 + d = 1.202171389 Solving -2.472222222 + d = 1.202171389 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '2.472222222' to each side of the equation. -2.472222222 + 2.472222222 + d = 1.202171389 + 2.472222222 Combine like terms: -2.472222222 + 2.472222222 = 0.000000000 0.000000000 + d = 1.202171389 + 2.472222222 d = 1.202171389 + 2.472222222 Combine like terms: 1.202171389 + 2.472222222 = 3.674393611 d = 3.674393611 Simplifying d = 3.674393611

Subproblem 2

d + -2.472222222 = -1.202171389 Simplifying d + -2.472222222 = -1.202171389 Reorder the terms: -2.472222222 + d = -1.202171389 Solving -2.472222222 + d = -1.202171389 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '2.472222222' to each side of the equation. -2.472222222 + 2.472222222 + d = -1.202171389 + 2.472222222 Combine like terms: -2.472222222 + 2.472222222 = 0.000000000 0.000000000 + d = -1.202171389 + 2.472222222 d = -1.202171389 + 2.472222222 Combine like terms: -1.202171389 + 2.472222222 = 1.270050833 d = 1.270050833 Simplifying d = 1.270050833

Solution

The solution to the problem is based on the solutions from the subproblems. d = {3.674393611, 1.270050833}

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