.5x+1/4x=13.5

Solution for .5x+1/4x=13.5 equation:

.5x+1/4x=13.5

We move all terms to the left:

.5x+1/4x-(13.5)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We add all the numbers together, and all the variables

.5x+1/4x-13.5=0

We multiply all the terms by the denominator


(.5x)*4x-(13.5)*4x+1=0

We add all the numbers together, and all the variables

(+.5x)*4x-(13.5)*4x+1=0

We multiply parentheses

4x^2-54x+1=0

a = 4; b = -54; c = +1;
Δ = b2-4ac
Δ = -542-4·4·1
Δ = 2900
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:

$\sqrt{\Delta}=\sqrt{2900}=\sqrt{100*29}=\sqrt{100}*\sqrt{29}=10\sqrt{29}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-54)-10\sqrt{29}}{2*4}=\frac{54-10\sqrt{29}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-54)+10\sqrt{29}}{2*4}=\frac{54+10\sqrt{29}}{8}
$