Product of solutions of quadratic equations equal constant term

Question

I recently saw the following reasoning:

Since $\alpha^2, \beta^2$ are roots of $X^2 - 8X + 25 = 0$ it follows: $$ \alpha^2\beta^2 = 25 $$

I don't understand why this is the case. I feel like I'm missing something obvious. Any help is welcome.

Answer 1

Due to the Root coefficient relationship, $\alpha^2\beta^2=25$.

Note that if $\alpha^2$ and $\beta^2$ are roots, then $(x-\alpha^2)(x-\beta^2)=0$

Where can you go from there?

Bonus question: What is $\alpha^2+\beta^2$?

Answer 2

If $\alpha^2$ and $\beta^2$ are the roots, then $X^2-8X+25=(X-\alpha^2)(X-\beta^2)=X^2-(\alpha^2+\beta^2)X+\alpha^2\beta^2$

Just equate the constant terms.

https://en.wikipedia.org/wiki/Vieta%27s_formulas

Answer 3

Note that for any quadratic equation $$ax^2 + bx +c =0$$

We have $$ x_1 x_2 = c/a $$

Thus , for your equation $$ x_1 x_2 = \alpha^2\beta^2=25$$