Question

One of the roots of the quadratic equation ax+bc+c is twice the other,show that 2b²=9ac

Answer 1

Let m and n be the two roots of the quadratic equation. 
and let m = 2n

we know that
    m + n = $\dfrac{-b}{a}$     and   m*n = $\dfrac{c}{a}$

m + n = $\dfrac{-b}{a}$ ⇒ 3n = $\dfrac{-b}{a}$         ------ 1

m*h = $\dfrac{c}{a}$  ⇒ 2$n^{2}$ = $\dfrac{c}{a}$     ------ 2

from eqn 1
  n = $\dfrac{-b}{3a}$   ⇒  $n^{2}$ = $\dfrac{b^{2}}{9a^{2}}$

substituting it in eqn 2

2 $\dfrac{b^{2}}{9a^{2}}$ = $\dfrac{c}{a}$

⇒ 2$b^{2}$ = 9ac