Question

If one root of the quadratic equation ax² + bx + c = 0 is double the other, prove that 2b² = 9 ac.

Answer 1

ax² + bx + c = 0.
Let the roots be α and 2α
Sum of roots = `(-b)/a`
⇒ α + 2α = `(-b)/a`
⇒ 3α = `(-b)/a`
⇒ α = `-(b)/(3a)`      ...(i)
Product of root = `c/a`
⇒ 2α² = `c/a`
α² = `c/(2a), α = sqrt(c/(2a)`  ...(iii)
Equation (i) = (ii)
`(-b)/(3a) = sqrt(c/(2a))   ...("squaring both side")`
`b^2/(9a^2) = c/(2a)`
⇒ 2b² = 9ac.
Hence Proved.