Question

If $x^{2}-ax-21=0$ and $x^{2}-3ax+35=0$ with $a>0$ have a common root, then $a$ equals:

• A. 1
• B. 2
• C. 4
• D. 5

Hint:

For two quadratics with a common root, subtract to eliminate $x^2$ and solve for the root in terms of parameters; then substitute back.

Answer 1

The Correct Option is C

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 $r^2-ar-21=0$: \[ \left(\frac{28}{a}\right)^2-a\left(\frac{28}{a}\right)-21=0 \;\Rightarrow\; \frac{784}{a^2}-49=0 \;\Rightarrow\; a^2=16 \;\Rightarrow\; a=4 \;(a>0). \]