If $x^{2}-ax-21=0$ and $x^{2}-3ax+35=0$ with $a>0$ have a common root, then $a$ equals:
5
Let the common root be $r$. Subtract the equations: \[ (-3a\,r+35)-(-a\,r-21)=0 \;\Rightarrow\; -2a r+56=0 \;\Rightarrow\; r=\frac{28}{a}. \] Plug into $r^2-ar-21=0$: \[ \left(\frac{28}{a}\right)^2-a\left(\frac{28}{a}\right)-21=0 \;\Rightarrow\; \frac{784}{a^2}-49=0 \;\Rightarrow\; a^2=16 \;\Rightarrow\; a=4 \;(a>0). \]
If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to .
Find the missing code:
L1#1O2~2, J2#2Q3~3, _______, F4#4U5~5, D5#5W6~6