T^2-3T-19=0

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Solution for T^2-3T-19=0 equation:


Simplifying
T2 + -3T + -19 = 0

Reorder the terms:
-19 + -3T + T2 = 0

Solving
-19 + -3T + T2 = 0

Solving for variable 'T'.

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + -3T + 19 + T2 = 0 + 19

Reorder the terms:
-19 + 19 + -3T + T2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + -3T + T2 = 0 + 19
-3T + T2 = 0 + 19

Combine like terms: 0 + 19 = 19
-3T + T2 = 19

The T term is -3T.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3T + 2.25 + T2 = 19 + 2.25

Reorder the terms:
2.25 + -3T + T2 = 19 + 2.25

Combine like terms: 19 + 2.25 = 21.25
2.25 + -3T + T2 = 21.25

Factor a perfect square on the left side:
(T + -1.5)(T + -1.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(T + -1.5) equal to 4.609772229 and -4.609772229.

Subproblem 1

T + -1.5 = 4.609772229 Simplifying T + -1.5 = 4.609772229 Reorder the terms: -1.5 + T = 4.609772229 Solving -1.5 + T = 4.609772229 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + T = 4.609772229 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + T = 4.609772229 + 1.5 T = 4.609772229 + 1.5 Combine like terms: 4.609772229 + 1.5 = 6.109772229 T = 6.109772229 Simplifying T = 6.109772229

Subproblem 2

T + -1.5 = -4.609772229 Simplifying T + -1.5 = -4.609772229 Reorder the terms: -1.5 + T = -4.609772229 Solving -1.5 + T = -4.609772229 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + T = -4.609772229 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + T = -4.609772229 + 1.5 T = -4.609772229 + 1.5 Combine like terms: -4.609772229 + 1.5 = -3.109772229 T = -3.109772229 Simplifying T = -3.109772229

Solution

The solution to the problem is based on the solutions from the subproblems. T = {6.109772229, -3.109772229}

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