3x=1/x-4x+10

Solution for 3x=1/x-4x+10 equation:

3x=1/x-4x+10

We move all terms to the left:

3x-(1/x-4x+10)=0


Domain of the equation: x-4x+10)!=0

x∈R


We add all the numbers together, and all the variables

3x-(-4x+1/x+10)=0

We get rid of parentheses

3x+4x-1/x-10=0

We multiply all the terms by the denominator


3x*x+4x*x-10*x-1=0

We add all the numbers together, and all the variables

-10x+3x*x+4x*x-1=0

Wy multiply elements

3x^2+4x^2-10x-1=0

We add all the numbers together, and all the variables

7x^2-10x-1=0

a = 7; b = -10; c = -1;
Δ = b2-4ac
Δ = -102-4·7·(-1)
Δ = 128
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:

$\sqrt{\Delta}=\sqrt{128}=\sqrt{64*2}=\sqrt{64}*\sqrt{2}=8\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-8\sqrt{2}}{2*7}=\frac{10-8\sqrt{2}}{14}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+8\sqrt{2}}{2*7}=\frac{10+8\sqrt{2}}{14}
$