f^2-22f-47=0

Solution for f^2-22f-47=0 equation:

f^2-22f-47=0

a = 1; b = -22; c = -47;
Δ = b2-4ac
Δ = -222-4·1·(-47)
Δ = 672
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$f_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$f_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{672}=\sqrt{16*42}=\sqrt{16}*\sqrt{42}=4\sqrt{42}$
$f_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-22)-4\sqrt{42}}{2*1}=\frac{22-4\sqrt{42}}{2}
$
$f_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-22)+4\sqrt{42}}{2*1}=\frac{22+4\sqrt{42}}{2}
$