Let $x_1,x_2 \in \mathbb{R}$ be the roots of the equation
$x^2+px+q=0$. Find $p$ and $q$ if it is known that $x_1+1$ and $x_2+1$
are the roots of the equation $x^2-p^2x+pq=0$.
The roots of $x^2+px+q=0$ satisfy $$x_1+x_2=-\dfrac{b}{a}=-p,x_1x_2=\dfrac{c}{a}=q.$$ Also the roots of $x^2-p^2x+pq=0$ satisfy $$x_1+x_2=x_1+1+x_2+1=x_1+x_2+2=-\dfrac{b}{a}=p^2,x_1x_2=(x_1+1)(x_2+1)=\dfrac{c}{a}=pq.$$ Is it confusing I am using $x_1$ and $x_2$ for the roots of both equations? How can I improve what I've written? What to do next?
Best Answer
Next substitute $x_1+x_2=-p$ into $x_1+x_2+2=p^2$ to obtain $2-p=p^2$, which you can solve for $p$.
To find $q$, expand $(x_1+1)(x_2+1)=pq$ to get $x_1x_2+(x_1+x_2)+1=pq$, which is the same as $q-p+1=pq$.
And you should not use $x_1,x_2$ to mean two different things.