Question

The equation is ax^2-18x+c=0. The sum of roots is 2 and the product of roots is 8/9. Find the coefficients of a and c.​

Answer 1

$\star\sf\ \ ax^2-18x+c=0\\ \\ \\ \bullet\sf \ \ \alpha+\beta=2\ \ \ ;\ \ \bullet\sf\ \ \alpha\beta= \ \large ^8\!/_9$

Now we know that

$\boxed{\sf\ \alpha+\beta= \dfrac{-Coefficient\ of x}{Coefficient \ of \ x^2}= \dfrac{-b}{a}}$

$\boxed{\sf\ \alpha\beta= \dfrac{Constant\ term }{Coefficient \ of \ x^2}= \dfrac{c}{a}}$

Now , compare the given equation by

$\sf\ \ ax^2+bc+c$

$\bullet\sf \ a= a\ \ ;\ \bullet\sf\ b= -18\ \ ;\ \bullet\sf\ c=c$

$\dashrightarrow\sf\ \alpha+\beta= \large ^{-b}\!/_a\\ \\ \\ \dashrightarrow\sf 2= \dfrac{-(-18)}{a}\\ \\ \\ \dashrightarrow\sf 2a=18\\ \\ \\ \dashrightarrow\sf a = \cancel{\dfrac{18}{2}}=9\\ \\ \\ \dashrightarrow\underline{\boxed{\sf\ a= 9 }}$

Now ,

$\dashrightarrow\sf\ \alpha\beta= \large ^{c}\!/_a\\ \\ \\ \dashrightarrow\sf \dfrac{8}{9}= \dfrac{c}{9}\\ \\ \\ \dashrightarrow\sf 8\times 9 =9\times c\\ \\ \\ \dashrightarrow\sf 72= 9c\\ \\ \\ \dashrightarrow\sf\ c= \cancel{\dfrac{72}{9}}\\ \\ \\ \dashrightarrow\underline{\boxed{\sf\ c= 8}}$