x+1/x+7=49

Solution for x+1/x+7=49 equation:

x+1/x+7=49

We move all terms to the left:

x+1/x+7-(49)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

x+1/x-42=0

We multiply all the terms by the denominator


x*x-42*x+1=0

We add all the numbers together, and all the variables

-42x+x*x+1=0

Wy multiply elements

x^2-42x+1=0

a = 1; b = -42; c = +1;
Δ = b2-4ac
Δ = -422-4·1·1
Δ = 1760
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:

$\sqrt{\Delta}=\sqrt{1760}=\sqrt{16*110}=\sqrt{16}*\sqrt{110}=4\sqrt{110}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-4\sqrt{110}}{2*1}=\frac{42-4\sqrt{110}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+4\sqrt{110}}{2*1}=\frac{42+4\sqrt{110}}{2}
$