90+(t+10.5)2t=180

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Solution for 90+(t+10.5)2t=180 equation:


Simplifying
90 + (t + 10.5) * 2t = 180

Reorder the terms:
90 + (10.5 + t) * 2t = 180

Reorder the terms for easier multiplication:
90 + 2t(10.5 + t) = 180
90 + (10.5 * 2t + t * 2t) = 180
90 + (21t + 2t2) = 180

Solving
90 + 21t + 2t2 = 180

Solving for variable 't'.

Reorder the terms:
90 + -180 + 21t + 2t2 = 180 + -180

Combine like terms: 90 + -180 = -90
-90 + 21t + 2t2 = 180 + -180

Combine like terms: 180 + -180 = 0
-90 + 21t + 2t2 = 0

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

Divide each side by '2'.
-45 + 10.5t + t2 = 0

Move the constant term to the right:

Add '45' to each side of the equation.
-45 + 10.5t + 45 + t2 = 0 + 45

Reorder the terms:
-45 + 45 + 10.5t + t2 = 0 + 45

Combine like terms: -45 + 45 = 0
0 + 10.5t + t2 = 0 + 45
10.5t + t2 = 0 + 45

Combine like terms: 0 + 45 = 45
10.5t + t2 = 45

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 = 45 + 27.5625

Reorder the terms:
27.5625 + 10.5t + t2 = 45 + 27.5625

Combine like terms: 45 + 27.5625 = 72.5625
27.5625 + 10.5t + t2 = 72.5625

Factor a perfect square on the left side:
(t + 5.25)(t + 5.25) = 72.5625

Calculate the square root of the right side: 8.518362519

Break this problem into two subproblems by setting 
(t + 5.25) equal to 8.518362519 and -8.518362519.

Subproblem 1

t + 5.25 = 8.518362519 Simplifying t + 5.25 = 8.518362519 Reorder the terms: 5.25 + t = 8.518362519 Solving 5.25 + t = 8.518362519 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 = 8.518362519 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + t = 8.518362519 + -5.25 t = 8.518362519 + -5.25 Combine like terms: 8.518362519 + -5.25 = 3.268362519 t = 3.268362519 Simplifying t = 3.268362519

Subproblem 2

t + 5.25 = -8.518362519 Simplifying t + 5.25 = -8.518362519 Reorder the terms: 5.25 + t = -8.518362519 Solving 5.25 + t = -8.518362519 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 = -8.518362519 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + t = -8.518362519 + -5.25 t = -8.518362519 + -5.25 Combine like terms: -8.518362519 + -5.25 = -13.768362519 t = -13.768362519 Simplifying t = -13.768362519

Solution

The solution to the problem is based on the solutions from the subproblems. t = {3.268362519, -13.768362519}

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