.75k+7=1/8k+27

Solution for .75k+7=1/8k+27 equation:

.75k+7=1/8k+27

We move all terms to the left:

.75k+7-(1/8k+27)=0


Domain of the equation: 8k+27)!=0

k∈R


We get rid of parentheses

.75k-1/8k-27+7=0

We multiply all the terms by the denominator


(.75k)*8k-27*8k+7*8k-1=0

We add all the numbers together, and all the variables

(+.75k)*8k-27*8k+7*8k-1=0

We multiply parentheses

8k^2-27*8k+7*8k-1=0

Wy multiply elements

8k^2-216k+56k-1=0

We add all the numbers together, and all the variables

8k^2-160k-1=0

a = 8; b = -160; c = -1;
Δ = b2-4ac
Δ = -1602-4·8·(-1)
Δ = 25632
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:

$\sqrt{\Delta}=\sqrt{25632}=\sqrt{144*178}=\sqrt{144}*\sqrt{178}=12\sqrt{178}$
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-160)-12\sqrt{178}}{2*8}=\frac{160-12\sqrt{178}}{16}
$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-160)+12\sqrt{178}}{2*8}=\frac{160+12\sqrt{178}}{16}
$