Let a and c be prime numbers and 'b' an integer Given that the quadratic equation ax^2+bx+c =0 has - Maths - Complex Numbers and Quadratic Equations

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

Koka Sri Lakshmi Divya Sai · Accepted Answer

Dear Student

Given equation ax2+bx+c=0Letb2-4ac=n2⇒b2-n2=4ac⇒b-nb+n=4acCaseI:b-n=4a and b+n=cCaseIIb-n=4c and b+n=aThese two cases are not possible as 2b=4a+c which is odd⇒b is not an integerCaseIII:b-n=2a and b+n=2c⇒b=a+cLet α,β be the two roots of the equation⇒αβ=ca and α+β=-ba=-a+ca=-1-ca⇒α=-1 and β=-ca

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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.