T(2t-3)/t+6;t=-2

Solution for T(2t-3)/t+6;t=-2 equation:

(2T-3)/T+6T=-2

We move all terms to the left:

(2T-3)/T+6T-(-2)=0


Domain of the equation: T!=0

T∈R


We add all the numbers together, and all the variables

6T+(2T-3)/T+2=0

We multiply all the terms by the denominator


6T*T+(2T-3)+2*T=0

We add all the numbers together, and all the variables

2T+6T*T+(2T-3)=0

Wy multiply elements

6T^2+2T+(2T-3)=0

We get rid of parentheses

6T^2+2T+2T-3=0

We add all the numbers together, and all the variables

6T^2+4T-3=0

a = 6; b = 4; c = -3;
Δ = b2-4ac
Δ = 42-4·6·(-3)
Δ = 88
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$T_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$T_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{88}=\sqrt{4*22}=\sqrt{4}*\sqrt{22}=2\sqrt{22}$
$T_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{22}}{2*6}=\frac{-4-2\sqrt{22}}{12}
$
$T_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{22}}{2*6}=\frac{-4+2\sqrt{22}}{12}
$