Answer:
[tex]-\frac{bc}{a^{2} }[/tex]
Step-by-step explanation:
We are given [tex]f(x)=ax^{2} +bx+c[/tex], [tex]\alpha[/tex] and [tex]\beta[/tex] are zeros of the function. We can use the sum and product of roots. You may have come across these equations before ↓
[tex]\alpha +\beta =-\frac{b}{a}[/tex]
[tex]\alpha \beta =\frac{c}{a}[/tex]
Since the coefficients are already in a, b, and c's, we do not need to sub in anything else.
Now, you are asked to evaluate [tex]\alpha ^{2} \beta +\alpha \beta ^{2}[/tex]. The next step after finding the roots above ↑, is to factorise this equation to be solved.
[tex]\alpha ^{2} \beta +\alpha \beta ^{2}[/tex]
= [tex]\alpha \beta (\alpha +\beta )[/tex]
Sub in each respective roots,
= [tex]\frac{c}{a} (-\frac{b}{a} )[/tex]
= [tex]-\frac{bc}{a^{2} }[/tex]
Hope this helped! Ask me if there's any part of the working you don't understand :)
