21d^2-640d+1728=0

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Solution for 21d^2-640d+1728=0 equation:


Simplifying
21d2 + -640d + 1728 = 0

Reorder the terms:
1728 + -640d + 21d2 = 0

Solving
1728 + -640d + 21d2 = 0

Solving for variable 'd'.

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

Divide each side by '21'.
82.28571429 + -30.47619048d + d2 = 0

Move the constant term to the right:

Add '-82.28571429' to each side of the equation.
82.28571429 + -30.47619048d + -82.28571429 + d2 = 0 + -82.28571429

Reorder the terms:
82.28571429 + -82.28571429 + -30.47619048d + d2 = 0 + -82.28571429

Combine like terms: 82.28571429 + -82.28571429 = 0.00000000
0.00000000 + -30.47619048d + d2 = 0 + -82.28571429
-30.47619048d + d2 = 0 + -82.28571429

Combine like terms: 0 + -82.28571429 = -82.28571429
-30.47619048d + d2 = -82.28571429

The d term is -30.47619048d.  Take half its coefficient (-15.23809524).
Square it (232.1995465) and add it to both sides.

Add '232.1995465' to each side of the equation.
-30.47619048d + 232.1995465 + d2 = -82.28571429 + 232.1995465

Reorder the terms:
232.1995465 + -30.47619048d + d2 = -82.28571429 + 232.1995465

Combine like terms: -82.28571429 + 232.1995465 = 149.91383221
232.1995465 + -30.47619048d + d2 = 149.91383221

Factor a perfect square on the left side:
(d + -15.23809524)(d + -15.23809524) = 149.91383221

Calculate the square root of the right side: 12.243930423

Break this problem into two subproblems by setting 
(d + -15.23809524) equal to 12.243930423 and -12.243930423.

Subproblem 1

d + -15.23809524 = 12.243930423 Simplifying d + -15.23809524 = 12.243930423 Reorder the terms: -15.23809524 + d = 12.243930423 Solving -15.23809524 + d = 12.243930423 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '15.23809524' to each side of the equation. -15.23809524 + 15.23809524 + d = 12.243930423 + 15.23809524 Combine like terms: -15.23809524 + 15.23809524 = 0.00000000 0.00000000 + d = 12.243930423 + 15.23809524 d = 12.243930423 + 15.23809524 Combine like terms: 12.243930423 + 15.23809524 = 27.482025663 d = 27.482025663 Simplifying d = 27.482025663

Subproblem 2

d + -15.23809524 = -12.243930423 Simplifying d + -15.23809524 = -12.243930423 Reorder the terms: -15.23809524 + d = -12.243930423 Solving -15.23809524 + d = -12.243930423 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '15.23809524' to each side of the equation. -15.23809524 + 15.23809524 + d = -12.243930423 + 15.23809524 Combine like terms: -15.23809524 + 15.23809524 = 0.00000000 0.00000000 + d = -12.243930423 + 15.23809524 d = -12.243930423 + 15.23809524 Combine like terms: -12.243930423 + 15.23809524 = 2.994164817 d = 2.994164817 Simplifying d = 2.994164817

Solution

The solution to the problem is based on the solutions from the subproblems. d = {27.482025663, 2.994164817}

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