2t(t+10.5)=180

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Solution for 2t(t+10.5)=180 equation:


Simplifying
2t(t + 10.5) = 180

Reorder the terms:
2t(10.5 + t) = 180
(10.5 * 2t + t * 2t) = 180
(21t + 2t2) = 180

Solving
21t + 2t2 = 180

Solving for variable 't'.

Reorder the terms:
-180 + 21t + 2t2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 21t + 2t2 = 0

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

Divide each side by '2'.
-90 + 10.5t + t2 = 0

Move the constant term to the right:

Add '90' to each side of the equation.
-90 + 10.5t + 90 + t2 = 0 + 90

Reorder the terms:
-90 + 90 + 10.5t + t2 = 0 + 90

Combine like terms: -90 + 90 = 0
0 + 10.5t + t2 = 0 + 90
10.5t + t2 = 0 + 90

Combine like terms: 0 + 90 = 90
10.5t + t2 = 90

The t term is 10.5t.  Take half its coefficient (5.25).
Square it (27.5625) and add it to both sides.

Add '27.5625' to each side of the equation.
10.5t + 27.5625 + t2 = 90 + 27.5625

Reorder the terms:
27.5625 + 10.5t + t2 = 90 + 27.5625

Combine like terms: 90 + 27.5625 = 117.5625
27.5625 + 10.5t + t2 = 117.5625

Factor a perfect square on the left side:
(t + 5.25)(t + 5.25) = 117.5625

Calculate the square root of the right side: 10.842624221

Break this problem into two subproblems by setting 
(t + 5.25) equal to 10.842624221 and -10.842624221.

Subproblem 1

t + 5.25 = 10.842624221 Simplifying t + 5.25 = 10.842624221 Reorder the terms: 5.25 + t = 10.842624221 Solving 5.25 + t = 10.842624221 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + t = 10.842624221 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + t = 10.842624221 + -5.25 t = 10.842624221 + -5.25 Combine like terms: 10.842624221 + -5.25 = 5.592624221 t = 5.592624221 Simplifying t = 5.592624221

Subproblem 2

t + 5.25 = -10.842624221 Simplifying t + 5.25 = -10.842624221 Reorder the terms: 5.25 + t = -10.842624221 Solving 5.25 + t = -10.842624221 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + t = -10.842624221 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + t = -10.842624221 + -5.25 t = -10.842624221 + -5.25 Combine like terms: -10.842624221 + -5.25 = -16.092624221 t = -16.092624221 Simplifying t = -16.092624221

Solution

The solution to the problem is based on the solutions from the subproblems. t = {5.592624221, -16.092624221}

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