(t-3)(t-6)=6

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Solution for (t-3)(t-6)=6 equation:


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

Reorder the terms:
(-3 + t)(t + -6) = 6

Reorder the terms:
(-3 + t)(-6 + t) = 6

Multiply (-3 + t) * (-6 + t)
(-3(-6 + t) + t(-6 + t)) = 6
((-6 * -3 + t * -3) + t(-6 + t)) = 6
((18 + -3t) + t(-6 + t)) = 6
(18 + -3t + (-6 * t + t * t)) = 6
(18 + -3t + (-6t + t2)) = 6

Combine like terms: -3t + -6t = -9t
(18 + -9t + t2) = 6

Solving
18 + -9t + t2 = 6

Solving for variable 't'.

Reorder the terms:
18 + -6 + -9t + t2 = 6 + -6

Combine like terms: 18 + -6 = 12
12 + -9t + t2 = 6 + -6

Combine like terms: 6 + -6 = 0
12 + -9t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-12' to each side of the equation.
12 + -9t + -12 + t2 = 0 + -12

Reorder the terms:
12 + -12 + -9t + t2 = 0 + -12

Combine like terms: 12 + -12 = 0
0 + -9t + t2 = 0 + -12
-9t + t2 = 0 + -12

Combine like terms: 0 + -12 = -12
-9t + t2 = -12

The t term is -9t.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9t + 20.25 + t2 = -12 + 20.25

Reorder the terms:
20.25 + -9t + t2 = -12 + 20.25

Combine like terms: -12 + 20.25 = 8.25
20.25 + -9t + t2 = 8.25

Factor a perfect square on the left side:
(t + -4.5)(t + -4.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(t + -4.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

t + -4.5 = 2.872281323 Simplifying t + -4.5 = 2.872281323 Reorder the terms: -4.5 + t = 2.872281323 Solving -4.5 + t = 2.872281323 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = 2.872281323 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = 2.872281323 + 4.5 t = 2.872281323 + 4.5 Combine like terms: 2.872281323 + 4.5 = 7.372281323 t = 7.372281323 Simplifying t = 7.372281323

Subproblem 2

t + -4.5 = -2.872281323 Simplifying t + -4.5 = -2.872281323 Reorder the terms: -4.5 + t = -2.872281323 Solving -4.5 + t = -2.872281323 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = -2.872281323 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = -2.872281323 + 4.5 t = -2.872281323 + 4.5 Combine like terms: -2.872281323 + 4.5 = 1.627718677 t = 1.627718677 Simplifying t = 1.627718677

Solution

The solution to the problem is based on the solutions from the subproblems. t = {7.372281323, 1.627718677}

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