Using the condition that the roots of ax2+bx+c=0 may be in the ratio m:n is mnb2=ac(m+n)2. (i) If the roots are α=β, then α⋅αb2‌=ac(α+α)2 ⇒‌‌b2‌=4ac (ii) If the roots are α=2β, then ‌β⋅2βb2‌=ac(β+2β)2 ⇒2b2‌=9ac (iii) If the roots are α=3β, then ‌β⋅3βb2‌=ac(β+3β)2 ⇒3b2‌=16ac (iv) If the roots are α=β2, then ‌(a2c)‌