Question

Let a and c be prime numbers and 'b' an integer. Given that the quadratic equation ax^2+bx+c =0 has rational roots, show tha one of the root is independent of the coefficients. Find the 2 roots.

Answer 1

Dear Student

$$\begin{gathered} Givenequation \\ a{x}^2+bx+c=0 \\ Let \\ {b}^2-4ac=n^2 \\ \Rightarrow b^2-n^2=4ac \\ \Rightarrow \left(b-n\right)\left(b+n\right)=4ac \\ CaseI: \\ b-n=4aandb+n=c \\ CaseII \\ b-n=4candb+n=a \\ Thesetwocasesarenotpossibleas2b=4a+cwhichisodd \\ \Rightarrow bisnotaninteger \\ CaseIII: \\ b-n=2aandb+n=2c \\ \Rightarrow b=a+c \\ Let\alpha ,\beta bethetworootsoftheequation \\ \Rightarrow \alpha \beta =\frac{c}{a}and \\ \alpha +\beta =\frac{-b}{a}=-\frac{a+c}{a}=-1-\frac{c}{a} \\ \Rightarrow \alpha =-1and\beta =-\frac{c}{a} \end{gathered}$$

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