Question

Find the equation whose roots are $2\alpha +1$ and $2\beta +1$, where $\alpha$ and $\beta$ are roots of $x^2-ax+b=0$

Answer 1

$\alpha +\beta =a,\alpha \beta =b$

let equation whose roots are $2\alpha +1,2\beta +1$

be $x^2+Ax+B=0$

then $-A=(2\alpha +1)+(2\beta +1)=0$

$-A=2\left(a\right)+2=2(a+1)$

$B=(2\alpha +1)(2\beta +1)$

$=4\alpha \beta +2\alpha +2\beta +1$

$=4b+2a+1$

$\therefore$ required equation is

$x^2-2(a+1)x+4b+2a+1=0$