## Related Roots

Suppose the equation
$z^2 +2z+5=0$
has roots
$\alpha , \; \beta$
.
For the equation above
$\alpha + \beta =-2, \; \alpha \beta =5$
.
What are the values of
$(2 \alpha +1)(2 \beta +1), \; (2 \alpha +1)+(2 \beta +1)$

$(2 \alpha +1), \; (2 \beta +1)$
are called related roots.
$(2 \alpha +1)(2 \beta +1)=4 \alpha \beta + 2( \alpha + \beta )+1 =4(5)+2(-2)+1=17$

$(2 \alpha +1)+(2 \beta +1)= 2( \alpha + \beta )+2 =2(-2)+2=-2$
.
The quadratic equation with roots equal to these related roots is
$z^2+2z+17=0$
.
Roots are related by a linear transformation. For it to be possible to write down an equation in this way, roots must be related by the same transformation.