12x+5=15+1/x

Solution for 12x+5=15+1/x equation:

12x+5=15+1/x

We move all terms to the left:

12x+5-(15+1/x)=0


Domain of the equation: x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

12x-(1/x+15)+5=0

We get rid of parentheses

12x-1/x-15+5=0

We multiply all the terms by the denominator


12x*x-15*x+5*x-1=0

We add all the numbers together, and all the variables

-10x+12x*x-1=0

Wy multiply elements

12x^2-10x-1=0

a = 12; b = -10; c = -1;
Δ = b2-4ac
Δ = -102-4·12·(-1)
Δ = 148
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{148}=\sqrt{4*37}=\sqrt{4}*\sqrt{37}=2\sqrt{37}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-2\sqrt{37}}{2*12}=\frac{10-2\sqrt{37}}{24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+2\sqrt{37}}{2*12}=\frac{10+2\sqrt{37}}{24}
$