If the equations 3x^2+mx+1=0 and 2x^2+nx+1=0 have ONE common root, prove that 2m^2+3n^2-5mn+1=0.

Question

It is an algebra qustion involving properties of a quadratic equation:

3x²+mx+1=0

2x²+nx+1=0

If these two equations share ONE (not two) common root, prove that:

2m²+3n²-5mn+1=0.

Answer 1

Let α be the common root.

3α²+mα+1=0 --> 3α²+mα=-1

2α²+nα+1= 0 --> 2α²+nα=-1

 

then  3α²+mα=2α²+nα --> α² +(m-n)α=0   -->

α[α+(m-n)]=0 so α=0  or α+(m-n)=0 --> α=n-m (but α=0 is not a root)

so α=n-m 

 

So substitute this into one of the 2 eqs: 

3(n-m)²+m(n-m)+1=0  --> 3(n²+m²-2mn)+mn-m²+1=0  -->

3n²+2m²-5mn+1=0