T(2T-3)=30

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Solution for T(2T-3)=30 equation:


Simplifying
T(2T + -3) = 30

Reorder the terms:
T(-3 + 2T) = 30
(-3 * T + 2T * T) = 30
(-3T + 2T2) = 30

Solving
-3T + 2T2 = 30

Solving for variable 'T'.

Reorder the terms:
-30 + -3T + 2T2 = 30 + -30

Combine like terms: 30 + -30 = 0
-30 + -3T + 2T2 = 0

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

Divide each side by '2'.
-15 + -1.5T + T2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + -1.5T + 15 + T2 = 0 + 15

Reorder the terms:
-15 + 15 + -1.5T + T2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + -1.5T + T2 = 0 + 15
-1.5T + T2 = 0 + 15

Combine like terms: 0 + 15 = 15
-1.5T + T2 = 15

The T term is -1.5T.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5T + 0.5625 + T2 = 15 + 0.5625

Reorder the terms:
0.5625 + -1.5T + T2 = 15 + 0.5625

Combine like terms: 15 + 0.5625 = 15.5625
0.5625 + -1.5T + T2 = 15.5625

Factor a perfect square on the left side:
(T + -0.75)(T + -0.75) = 15.5625

Calculate the square root of the right side: 3.94493346

Break this problem into two subproblems by setting 
(T + -0.75) equal to 3.94493346 and -3.94493346.

Subproblem 1

T + -0.75 = 3.94493346 Simplifying T + -0.75 = 3.94493346 Reorder the terms: -0.75 + T = 3.94493346 Solving -0.75 + T = 3.94493346 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + T = 3.94493346 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + T = 3.94493346 + 0.75 T = 3.94493346 + 0.75 Combine like terms: 3.94493346 + 0.75 = 4.69493346 T = 4.69493346 Simplifying T = 4.69493346

Subproblem 2

T + -0.75 = -3.94493346 Simplifying T + -0.75 = -3.94493346 Reorder the terms: -0.75 + T = -3.94493346 Solving -0.75 + T = -3.94493346 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + T = -3.94493346 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + T = -3.94493346 + 0.75 T = -3.94493346 + 0.75 Combine like terms: -3.94493346 + 0.75 = -3.19493346 T = -3.19493346 Simplifying T = -3.19493346

Solution

The solution to the problem is based on the solutions from the subproblems. T = {4.69493346, -3.19493346}

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