Alpha and Beta question [Addmath, quadratic equations]

Question

Please help.

Question: Addmath (Quadratic Equations)

Given $\alpha$ and $\beta$ are the roots of the quadratic equation $2x^2 - 6x + 5 = 0$, form an quadratic equation with the roots $\alpha + 1$ and $\beta + 1$.

Answer 1

$$x^2 - 3x + \frac 52 = (x - \alpha)(x - \beta) = x^2 - (\alpha + \beta)x + \alpha \beta$$

Then $\alpha + \beta = 3$ and $\alpha \beta = \frac 52$. So $$(x-(\alpha + 1))(x - (\beta + 1)) = x^2 - (\alpha + \beta+2)x +(\alpha \beta + \alpha + \beta + 1)$$

You should be able to do the rest.

Answer 2

To add $1$ to each root, simply shift the function right $1$ unit by replacing $x$ with $x-1$. Then you get \begin{gather} 2(x-1)^2-6(x-1)+5=0\\ 2(x^2-2x+1)-6(x-1)+5=0\\ 2x^2-10x+13=0 \end{gather}