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Algebra Level pending

Let there be three quadratic equations such as: $x^2-2rp_rx+r=0 , r=1,2,3$

Each pair of these quadratic equations have a common root (exactly one). Find the common root of the equation with $r=1$ and $r=3$ .

\sqrt{\frac{3}{2}} or -\sqrt{\frac{3}{2}}

0 1

\sqrt{\frac{1}{2}} or -\sqrt{\frac{1}{2}}

\frac{1}{2}

\frac{1}{3}

1 solution

Gautam Sharma
Apr 9, 2015

Let the roots of the three equations be $\alpha,\beta$ ; $\beta,\gamma$ ; $\alpha,\gamma$

Now from r=1 eq;

$\alpha \beta =1$ ............(1)

Now from r=2 eq;

$\gamma \beta =2$ .........(2)

Now from r=3 eq;

$\gamma \alpha =3$ ...........(3)

Multiply eq 1 and 3:

$\alpha ^2 \gamma \beta=3$

But $\gamma \beta =2$

Hence $\alpha=\sqrt{\frac{3}{2}}$ or $-\sqrt{\frac{3}{2}}$