(2t)(t-6)=(18)(t-6)+3

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Solution for (2t)(t-6)=(18)(t-6)+3 equation:


Simplifying
(2t)(t + -6) = (18)(t + -6) + 3

Remove parenthesis around (2t)
2t(t + -6) = (18)(t + -6) + 3

Reorder the terms:
2t(-6 + t) = (18)(t + -6) + 3
(-6 * 2t + t * 2t) = (18)(t + -6) + 3
(-12t + 2t2) = (18)(t + -6) + 3

Reorder the terms:
-12t + 2t2 = 18(-6 + t) + 3
-12t + 2t2 = (-6 * 18 + t * 18) + 3
-12t + 2t2 = (-108 + 18t) + 3

Reorder the terms:
-12t + 2t2 = -108 + 3 + 18t

Combine like terms: -108 + 3 = -105
-12t + 2t2 = -105 + 18t

Solving
-12t + 2t2 = -105 + 18t

Solving for variable 't'.

Reorder the terms:
105 + -12t + -18t + 2t2 = -105 + 18t + 105 + -18t

Combine like terms: -12t + -18t = -30t
105 + -30t + 2t2 = -105 + 18t + 105 + -18t

Reorder the terms:
105 + -30t + 2t2 = -105 + 105 + 18t + -18t

Combine like terms: -105 + 105 = 0
105 + -30t + 2t2 = 0 + 18t + -18t
105 + -30t + 2t2 = 18t + -18t

Combine like terms: 18t + -18t = 0
105 + -30t + 2t2 = 0

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

Divide each side by '2'.
52.5 + -15t + t2 = 0

Move the constant term to the right:

Add '-52.5' to each side of the equation.
52.5 + -15t + -52.5 + t2 = 0 + -52.5

Reorder the terms:
52.5 + -52.5 + -15t + t2 = 0 + -52.5

Combine like terms: 52.5 + -52.5 = 0.0
0.0 + -15t + t2 = 0 + -52.5
-15t + t2 = 0 + -52.5

Combine like terms: 0 + -52.5 = -52.5
-15t + t2 = -52.5

The t term is -15t.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15t + 56.25 + t2 = -52.5 + 56.25

Reorder the terms:
56.25 + -15t + t2 = -52.5 + 56.25

Combine like terms: -52.5 + 56.25 = 3.75
56.25 + -15t + t2 = 3.75

Factor a perfect square on the left side:
(t + -7.5)(t + -7.5) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(t + -7.5) equal to 1.936491673 and -1.936491673.

Subproblem 1

t + -7.5 = 1.936491673 Simplifying t + -7.5 = 1.936491673 Reorder the terms: -7.5 + t = 1.936491673 Solving -7.5 + t = 1.936491673 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + t = 1.936491673 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = 1.936491673 + 7.5 t = 1.936491673 + 7.5 Combine like terms: 1.936491673 + 7.5 = 9.436491673 t = 9.436491673 Simplifying t = 9.436491673

Subproblem 2

t + -7.5 = -1.936491673 Simplifying t + -7.5 = -1.936491673 Reorder the terms: -7.5 + t = -1.936491673 Solving -7.5 + t = -1.936491673 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + t = -1.936491673 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = -1.936491673 + 7.5 t = -1.936491673 + 7.5 Combine like terms: -1.936491673 + 7.5 = 5.563508327 t = 5.563508327 Simplifying t = 5.563508327

Solution

The solution to the problem is based on the solutions from the subproblems. t = {9.436491673, 5.563508327}

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