Relationship between roots and equations

Question

I'm stuck on topic of relationship between roots and equations. The roots of $x^2 -2x +3 =0$, are $\alpha$ and $\beta$. Find the equation whose roots are :

1- $\alpha+2$, $\beta+2$

2- $\alpha^2$, $\beta^2$

I know what the basis are like $\alpha+\beta$ and $\alpha\beta$ but I couldn't proceed further into it. Please help, my a-levels exams are near. Thanks in advance

Answer 1

Hint: Use the fact that the polynomial

$$p(x) = (x-r_1)(x-r_2)$$

has roots $r_1$ and $r_2$.

Answer 2

You must establish a relation between $s'=(\alpha+2)+(\beta+2)$, $p'=(\alpha+2)(\beta+2)$ and $s=\alpha+\beta$, $p=\alpha\beta$. Then the equation you're after is simply: $$x^2-s'x+p'=0.$$

Answer 3

The sum $S=\alpha+\beta=\frac{-(-2)}{1}=2$ and the product $P=\alpha\beta=\frac{3}{1}=3$

now

1) $\alpha+2+\beta+2=S+4=6$ and $(\alpha+2)(\beta+2)=P+2S+4=11$ so in this case the quadratic is $X^2-6X+11=0$

2) $\alpha^2+\beta^2=S^2-2P=-2$ and $\alpha^2\beta^2=P^2=9$ so here the quadratic is $X^2+2X+9=0$