2x(x-3)+1=7

Solution for 2x(x-3)+1=7 equation:

2x(x-3)+1=7

We move all terms to the left:

2x(x-3)+1-(7)=0

We add all the numbers together, and all the variables

2x(x-3)-6=0

We multiply parentheses

2x^2-6x-6=0

a = 2; b = -6; c = -6;
Δ = b2-4ac
Δ = -62-4·2·(-6)
Δ = 84
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:

$\sqrt{\Delta}=\sqrt{84}=\sqrt{4*21}=\sqrt{4}*\sqrt{21}=2\sqrt{21}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{21}}{2*2}=\frac{6-2\sqrt{21}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{21}}{2*2}=\frac{6+2\sqrt{21}}{4}
$