Find the base numeric system Find the numeric base we are using if $ x = 4 $ and $ x = 7 $ are zeros of $ 5x^2 - 50x + 118. $
When I plug in $ x = 4 $ and $ x = 7, $ I receive $ -2 $ and $ 13, $ respectively, but how do I proceed from that?

Here's a different way to do it.
$f (x)=5x^2 - 50_bx +118_b =$
$5x (x-b)+118_b $
$f (4)=20 (4-b)  +118_b$
$f (7)=35 (7-b)+118_b $ 
So $118_b= 20 (b-4)=35 (b-7) $
So $15b =7*35-80$
$b=11$.
Or we could note $35|118_b $ and $20|118_b $ so $140|118_b $ and $7|b-4$ and $4|b-7$.  If $140 =118_b $ then $b=11$ is a good guess.
And here's a third way:
$5*4^2 - 4*(5b+0) + (b^2+b+8)=0$
$b^2 -19b +88=0$
$b= \frac {19 \pm \sqrt {19^2- 4*88}}{2}= 8;11$
$5*7^2 -7*5b+(b^2+b+8)=0$
$b^2 -34b+253=0$
$b= \frac {34 \pm \sqrt {34^2- 4*253}}{2}= 11;23$
So $b=11$
Still my favorite way is $5 (x-4)(x-7) =5x^2-55x+140= 5x^2 -50_bx + 118_b$ So $50_b=5b=55$ and $118_b =b^2+b+8=140$ so $b=11$.
