8x+31.5=1/2x+1.5

Solution for 8x+31.5=1/2x+1.5 equation:

8x+31.5=1/2x+1.5

We move all terms to the left:

8x+31.5-(1/2x+1.5)=0


Domain of the equation: 2x+1.5)!=0

x∈R


We get rid of parentheses

8x-1/2x-1.5+31.5=0

We multiply all the terms by the denominator


8x*2x-(1.5)*2x+(31.5)*2x-1=0

We multiply parentheses

8x*2x-3x+63x-1=0

Wy multiply elements

16x^2-3x+63x-1=0

We add all the numbers together, and all the variables

16x^2+60x-1=0

a = 16; b = 60; c = -1;
Δ = b2-4ac
Δ = 602-4·16·(-1)
Δ = 3664
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:

$\sqrt{\Delta}=\sqrt{3664}=\sqrt{16*229}=\sqrt{16}*\sqrt{229}=4\sqrt{229}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(60)-4\sqrt{229}}{2*16}=\frac{-60-4\sqrt{229}}{32}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(60)+4\sqrt{229}}{2*16}=\frac{-60+4\sqrt{229}}{32}
$