Question

Find the condition that one root of ax² + bx + c = 0 may be four times the other.

Answer 1

Let the roots be α and 4α. Then,

α + 4α = 5α = \(-\frac{b}{a}\)

α . 4α = 4α² = \(\frac{c}{a}\)

From (i) α =  \(\frac{b}{5a}\)

∴ From (ii) 4. \(\big(-\frac{b}{5a}\big)^2\) = \(\frac{c}{a}\) ⇒ \(\frac{4b^2}{25a^2}\) = \(\frac{c}{a}\) ⇒ 4b² = 25ac.