3/4x-5=x+2

Solution for 3/4x-5=x+2 equation:

3/4x-5=x+2

We move all terms to the left:

3/4x-5-(x+2)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We get rid of parentheses

3/4x-x-2-5=0

We multiply all the terms by the denominator


-x*4x-2*4x-5*4x+3=0

Wy multiply elements

-4x^2-8x-20x+3=0

We add all the numbers together, and all the variables

-4x^2-28x+3=0

a = -4; b = -28; c = +3;
Δ = b2-4ac
Δ = -282-4·(-4)·3
Δ = 832
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:

$\sqrt{\Delta}=\sqrt{832}=\sqrt{64*13}=\sqrt{64}*\sqrt{13}=8\sqrt{13}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-28)-8\sqrt{13}}{2*-4}=\frac{28-8\sqrt{13}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-28)+8\sqrt{13}}{2*-4}=\frac{28+8\sqrt{13}}{-8}
$