39.6=8.4T-0.3T^2

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Solution for 39.6=8.4T-0.3T^2 equation:


Simplifying
39.6 = 8.4T + -0.3T2

Solving
39.6 = 8.4T + -0.3T2

Solving for variable 'T'.

Reorder the terms:
39.6 + -8.4T + 0.3T2 = 8.4T + -8.4T + -0.3T2 + 0.3T2

Combine like terms: 8.4T + -8.4T = 0.0
39.6 + -8.4T + 0.3T2 = 0.0 + -0.3T2 + 0.3T2
39.6 + -8.4T + 0.3T2 = -0.3T2 + 0.3T2

Combine like terms: -0.3T2 + 0.3T2 = 0.0
39.6 + -8.4T + 0.3T2 = 0.0

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

Divide each side by '0.3'.
132 + -28T + T2 = 0

Move the constant term to the right:

Add '-132' to each side of the equation.
132 + -28T + -132 + T2 = 0 + -132

Reorder the terms:
132 + -132 + -28T + T2 = 0 + -132

Combine like terms: 132 + -132 = 0
0 + -28T + T2 = 0 + -132
-28T + T2 = 0 + -132

Combine like terms: 0 + -132 = -132
-28T + T2 = -132

The T term is -28T.  Take half its coefficient (-14).
Square it (196) and add it to both sides.

Add '196' to each side of the equation.
-28T + 196 + T2 = -132 + 196

Reorder the terms:
196 + -28T + T2 = -132 + 196

Combine like terms: -132 + 196 = 64
196 + -28T + T2 = 64

Factor a perfect square on the left side:
(T + -14)(T + -14) = 64

Calculate the square root of the right side: 8

Break this problem into two subproblems by setting 
(T + -14) equal to 8 and -8.

Subproblem 1

T + -14 = 8 Simplifying T + -14 = 8 Reorder the terms: -14 + T = 8 Solving -14 + T = 8 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '14' to each side of the equation. -14 + 14 + T = 8 + 14 Combine like terms: -14 + 14 = 0 0 + T = 8 + 14 T = 8 + 14 Combine like terms: 8 + 14 = 22 T = 22 Simplifying T = 22

Subproblem 2

T + -14 = -8 Simplifying T + -14 = -8 Reorder the terms: -14 + T = -8 Solving -14 + T = -8 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '14' to each side of the equation. -14 + 14 + T = -8 + 14 Combine like terms: -14 + 14 = 0 0 + T = -8 + 14 T = -8 + 14 Combine like terms: -8 + 14 = 6 T = 6 Simplifying T = 6

Solution

The solution to the problem is based on the solutions from the subproblems. T = {22, 6}

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