10x+1/2x=65

Solution for 10x+1/2x=65 equation:

10x+1/2x=65

We move all terms to the left:

10x+1/2x-(65)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We multiply all the terms by the denominator


10x*2x-65*2x+1=0

Wy multiply elements

20x^2-130x+1=0

a = 20; b = -130; c = +1;
Δ = b2-4ac
Δ = -1302-4·20·1
Δ = 16820
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:

$\sqrt{\Delta}=\sqrt{16820}=\sqrt{3364*5}=\sqrt{3364}*\sqrt{5}=58\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-130)-58\sqrt{5}}{2*20}=\frac{130-58\sqrt{5}}{40}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-130)+58\sqrt{5}}{2*20}=\frac{130+58\sqrt{5}}{40}
$