3x-53/x=14

Solution for 3x-53/x=14 equation:

3x-53/x=14

We move all terms to the left:

3x-53/x-(14)=0


Domain of the equation: x!=0

x∈R


We multiply all the terms by the denominator


3x*x-14*x-53=0

We add all the numbers together, and all the variables

-14x+3x*x-53=0

Wy multiply elements

3x^2-14x-53=0

a = 3; b = -14; c = -53;
Δ = b2-4ac
Δ = -142-4·3·(-53)
Δ = 832
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:

$\sqrt{\Delta}=\sqrt{832}=\sqrt{64*13}=\sqrt{64}*\sqrt{13}=8\sqrt{13}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-8\sqrt{13}}{2*3}=\frac{14-8\sqrt{13}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+8\sqrt{13}}{2*3}=\frac{14+8\sqrt{13}}{6}
$