10=25-5T-T^2

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Solution for 10=25-5T-T^2 equation:


Simplifying
10 = 25 + -5T + -1T2

Solving
10 = 25 + -5T + -1T2

Solving for variable 'T'.

Combine like terms: 10 + -25 = -15
-15 + 5T + T2 = 25 + -5T + -1T2 + -25 + 5T + T2

Reorder the terms:
-15 + 5T + T2 = 25 + -25 + -5T + 5T + -1T2 + T2

Combine like terms: 25 + -25 = 0
-15 + 5T + T2 = 0 + -5T + 5T + -1T2 + T2
-15 + 5T + T2 = -5T + 5T + -1T2 + T2

Combine like terms: -5T + 5T = 0
-15 + 5T + T2 = 0 + -1T2 + T2
-15 + 5T + T2 = -1T2 + T2

Combine like terms: -1T2 + T2 = 0
-15 + 5T + T2 = 0

Begin completing the square.

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 5T + 15 + T2 = 0 + 15

Reorder the terms:
-15 + 15 + 5T + T2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 5T + T2 = 0 + 15
5T + T2 = 0 + 15

Combine like terms: 0 + 15 = 15
5T + T2 = 15

The T term is 5T.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5T + 6.25 + T2 = 15 + 6.25

Reorder the terms:
6.25 + 5T + T2 = 15 + 6.25

Combine like terms: 15 + 6.25 = 21.25
6.25 + 5T + T2 = 21.25

Factor a perfect square on the left side:
(T + 2.5)(T + 2.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(T + 2.5) equal to 4.609772229 and -4.609772229.

Subproblem 1

T + 2.5 = 4.609772229 Simplifying T + 2.5 = 4.609772229 Reorder the terms: 2.5 + T = 4.609772229 Solving 2.5 + T = 4.609772229 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + T = 4.609772229 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + T = 4.609772229 + -2.5 T = 4.609772229 + -2.5 Combine like terms: 4.609772229 + -2.5 = 2.109772229 T = 2.109772229 Simplifying T = 2.109772229

Subproblem 2

T + 2.5 = -4.609772229 Simplifying T + 2.5 = -4.609772229 Reorder the terms: 2.5 + T = -4.609772229 Solving 2.5 + T = -4.609772229 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + T = -4.609772229 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + T = -4.609772229 + -2.5 T = -4.609772229 + -2.5 Combine like terms: -4.609772229 + -2.5 = -7.109772229 T = -7.109772229 Simplifying T = -7.109772229

Solution

The solution to the problem is based on the solutions from the subproblems. T = {2.109772229, -7.109772229}

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