Question

Given that one root of the quadratic equation ax² + bx + c = 0 is three times the other, show that 3b² – 16ac.

Answer 1

The given quadratic equation is
ax² + bx + c = 0
Let α be the one root
Then other root = 3α
Now, Sum of the root = `(-b)/a`
⇒ α + 3α = `(-b)/a`
⇒ 4α = `(-b)/a`
⇒ α = `(-b)/(4a)`    ...(i)
Also product of the root α x 3α
= `c/a`
= 3α² = `c/a`
From equation (i) `3(-b/(4a))^2 = c/a`
⇒ `3 xx b^2/(16a^2) = c/a`
⇒ 3b² = 16ac Proved.