k+19=1/2k+59

Solution for k+19=1/2k+59 equation:

k+19=1/2k+59

We move all terms to the left:

k+19-(1/2k+59)=0


Domain of the equation: 2k+59)!=0

k∈R


We get rid of parentheses

k-1/2k-59+19=0

We multiply all the terms by the denominator


k*2k-59*2k+19*2k-1=0

Wy multiply elements

2k^2-118k+38k-1=0

We add all the numbers together, and all the variables

2k^2-80k-1=0

a = 2; b = -80; c = -1;
Δ = b2-4ac
Δ = -802-4·2·(-1)
Δ = 6408
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{6408}=\sqrt{36*178}=\sqrt{36}*\sqrt{178}=6\sqrt{178}$
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-80)-6\sqrt{178}}{2*2}=\frac{80-6\sqrt{178}}{4}
$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-80)+6\sqrt{178}}{2*2}=\frac{80+6\sqrt{178}}{4}
$