1/x=x-4

Solution for 1/x=x-4 equation:

1/x=x-4

We move all terms to the left:

1/x-(x-4)=0


Domain of the equation: x!=0

x∈R


We get rid of parentheses

1/x-x+4=0

We multiply all the terms by the denominator


-x*x+4*x+1=0

We add all the numbers together, and all the variables

4x-x*x+1=0

Wy multiply elements

-1x^2+4x+1=0

a = -1; b = 4; c = +1;
Δ = b2-4ac
Δ = 42-4·(-1)·1
Δ = 20
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:

$\sqrt{\Delta}=\sqrt{20}=\sqrt{4*5}=\sqrt{4}*\sqrt{5}=2\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{5}}{2*-1}=\frac{-4-2\sqrt{5}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{5}}{2*-1}=\frac{-4+2\sqrt{5}}{-2}
$