(a-2)+(a+4)/(a-1)=14

Solution for (a-2)+(a+4)/(a-1)=14 equation:

(a-2)+(a+4)/(a-1)=14

We move all terms to the left:

(a-2)+(a+4)/(a-1)-(14)=0


Domain of the equation: (a-1)!=0

We move all terms containing a to the left, all other terms to the right

a!=1

a∈R


We get rid of parentheses

a+(a+4)/(a-1)-2-14=0

We multiply all the terms by the denominator


a*(a-1)+(a+4)-2*(a-1)-14*(a-1)=0

We multiply parentheses

a^2-1a+(a+4)-2a-14a+2+14=0

We get rid of parentheses

a^2-1a+a-2a-14a+4+2+14=0

We add all the numbers together, and all the variables

a^2-16a+20=0

a = 1; b = -16; c = +20;
Δ = b2-4ac
Δ = -162-4·1·20
Δ = 176
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{176}=\sqrt{16*11}=\sqrt{16}*\sqrt{11}=4\sqrt{11}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-16)-4\sqrt{11}}{2*1}=\frac{16-4\sqrt{11}}{2}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-16)+4\sqrt{11}}{2*1}=\frac{16+4\sqrt{11}}{2}
$