(250)(4.2)(98-T)(0.9)(98-T)=(85)(4.2)(T-5)

Solution for (250)(4.2)(98-T)(0.9)(98-T)=(85)(4.2)(T-5) equation:

Simplifying
(250)(4.2)(98 + -1T)(0.9)(98 + -1T) = (85)(4.2)(T + -5)

Reorder the terms for easier multiplication:
250 * 4.2 * 0.9(98 + -1T)(98 + -1T) = (85)(4.2)(T + -5)

Multiply 250 * 4.2
1050 * 0.9(98 + -1T)(98 + -1T) = (85)(4.2)(T + -5)

Multiply 1050 * 0.9
945(98 + -1T)(98 + -1T) = (85)(4.2)(T + -5)

Multiply (98 + -1T) * (98 + -1T)
945(98(98 + -1T) + -1T * (98 + -1T)) = (85)(4.2)(T + -5)
945((98 * 98 + -1T * 98) + -1T * (98 + -1T)) = (85)(4.2)(T + -5)
945((9604 + -98T) + -1T * (98 + -1T)) = (85)(4.2)(T + -5)
945(9604 + -98T + (98 * -1T + -1T * -1T)) = (85)(4.2)(T + -5)
945(9604 + -98T + (-98T + 1T2)) = (85)(4.2)(T + -5)

Combine like terms: -98T + -98T = -196T
945(9604 + -196T + 1T2) = (85)(4.2)(T + -5)
(9604 * 945 + -196T * 945 + 1T2 * 945) = (85)(4.2)(T + -5)
(9075780 + -185220T + 945T2) = (85)(4.2)(T + -5)

Reorder the terms:
9075780 + -185220T + 945T2 = 85 * 4.2(-5 + T)

Multiply 85 * 4.2
9075780 + -185220T + 945T2 = 357(-5 + T)
9075780 + -185220T + 945T2 = (-5 * 357 + T * 357)
9075780 + -185220T + 945T2 = (-1785 + 357T)

Solving
9075780 + -185220T + 945T2 = -1785 + 357T

Solving for variable 'T'.

Reorder the terms:
9075780 + 1785 + -185220T + -357T + 945T2 = -1785 + 357T + 1785 + -357T

Combine like terms: 9075780 + 1785 = 9077565
9077565 + -185220T + -357T + 945T2 = -1785 + 357T + 1785 + -357T

Combine like terms: -185220T + -357T = -185577T
9077565 + -185577T + 945T2 = -1785 + 357T + 1785 + -357T

Reorder the terms:
9077565 + -185577T + 945T2 = -1785 + 1785 + 357T + -357T

Combine like terms: -1785 + 1785 = 0
9077565 + -185577T + 945T2 = 0 + 357T + -357T
9077565 + -185577T + 945T2 = 357T + -357T

Combine like terms: 357T + -357T = 0
9077565 + -185577T + 945T2 = 0

Factor out the Greatest Common Factor (GCF), '21'.
21(432265 + -8837T + 45T2) = 0

Ignore the factor 21.
Subproblem 1
Set the factor '(432265 + -8837T + 45T2)' equal to zero and attempt to solve:

Simplifying
432265 + -8837T + 45T2 = 0

Solving
432265 + -8837T + 45T2 = 0

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

Divide each side by '45'.
9605.888889 + -196.3777778T + T2 = 0

Move the constant term to the right:

Add '-9605.888889' to each side of the equation.
9605.888889 + -196.3777778T + -9605.888889 + T2 = 0 + -9605.888889

Reorder the terms:
9605.888889 + -9605.888889 + -196.3777778T + T2 = 0 + -9605.888889

Combine like terms: 9605.888889 + -9605.888889 = 0.000000
0.000000 + -196.3777778T + T2 = 0 + -9605.888889
-196.3777778T + T2 = 0 + -9605.888889

Combine like terms: 0 + -9605.888889 = -9605.888889
-196.3777778T + T2 = -9605.888889

The T term is -196.3777778T.  Take half its coefficient (-98.1888889).
Square it (9641.057903) and add it to both sides.

Add '9641.057903' to each side of the equation.
-196.3777778T + 9641.057903 + T2 = -9605.888889 + 9641.057903

Reorder the terms:
9641.057903 + -196.3777778T + T2 = -9605.888889 + 9641.057903

Combine like terms: -9605.888889 + 9641.057903 = 35.169014
9641.057903 + -196.3777778T + T2 = 35.169014

Factor a perfect square on the left side:
(T + -98.1888889)(T + -98.1888889) = 35.169014

Calculate the square root of the right side: 5.93034687

Break this problem into two subproblems by setting 
(T + -98.1888889) equal to 5.93034687 and -5.93034687.

Subproblem 1
T + -98.1888889 = 5.93034687

Simplifying
T + -98.1888889 = 5.93034687

Reorder the terms:
-98.1888889 + T = 5.93034687

Solving
-98.1888889 + T = 5.93034687

Solving for variable 'T'.

Move all terms containing T to the left, all other terms to the right.

Add '98.1888889' to each side of the equation.
-98.1888889 + 98.1888889 + T = 5.93034687 + 98.1888889

Combine like terms: -98.1888889 + 98.1888889 = 0.0000000
0.0000000 + T = 5.93034687 + 98.1888889
T = 5.93034687 + 98.1888889

Combine like terms: 5.93034687 + 98.1888889 = 104.11923577
T = 104.11923577

Simplifying
T = 104.11923577

Subproblem 2
T + -98.1888889 = -5.93034687

Simplifying
T + -98.1888889 = -5.93034687

Reorder the terms:
-98.1888889 + T = -5.93034687

Solving
-98.1888889 + T = -5.93034687

Solving for variable 'T'.

Move all terms containing T to the left, all other terms to the right.

Add '98.1888889' to each side of the equation.
-98.1888889 + 98.1888889 + T = -5.93034687 + 98.1888889

Combine like terms: -98.1888889 + 98.1888889 = 0.0000000
0.0000000 + T = -5.93034687 + 98.1888889
T = -5.93034687 + 98.1888889

Combine like terms: -5.93034687 + 98.1888889 = 92.25854203
T = 92.25854203

Simplifying
T = 92.25854203

Solution
The solution to the problem is based on the solutions
from the subproblems.
T = {104.11923577, 92.25854203}
Solution
T = {104.11923577, 92.25854203}