x+(x+1)+(x+2)/x+3=51

Solution for x+(x+1)+(x+2)/x+3=51 equation:

x+(x+1)+(x+2)/x+3=51

We move all terms to the left:

x+(x+1)+(x+2)/x+3-(51)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

x+(x+1)+(x+2)/x-48=0

We get rid of parentheses

x+x+(x+2)/x+1-48=0

We multiply all the terms by the denominator


x*x+x*x+(x+2)+1*x-48*x=0

We add all the numbers together, and all the variables

-47x+x*x+x*x+(x+2)=0

Wy multiply elements

x^2+x^2-47x+(x+2)=0

We get rid of parentheses

x^2+x^2-47x+x+2=0

We add all the numbers together, and all the variables

2x^2-46x+2=0

a = 2; b = -46; c = +2;
Δ = b2-4ac
Δ = -462-4·2·2
Δ = 2100
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:

$\sqrt{\Delta}=\sqrt{2100}=\sqrt{100*21}=\sqrt{100}*\sqrt{21}=10\sqrt{21}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-46)-10\sqrt{21}}{2*2}=\frac{46-10\sqrt{21}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-46)+10\sqrt{21}}{2*2}=\frac{46+10\sqrt{21}}{4}
$