7v^2-4v-1=0

Solution for 7v^2-4v-1=0 equation:

7v^2-4v-1=0

a = 7; b = -4; c = -1;
Δ = b2-4ac
Δ = -42-4·7·(-1)
Δ = 44
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$v_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$v_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{44}=\sqrt{4*11}=\sqrt{4}*\sqrt{11}=2\sqrt{11}$
$v_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{11}}{2*7}=\frac{4-2\sqrt{11}}{14}
$
$v_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{11}}{2*7}=\frac{4+2\sqrt{11}}{14}
$