Question

Find the condition that one root of the quadratic equation ax2+bx+c=0 shall be n times the other,where n is a positive integer.

Answer 1

Answer:

The required condition is

$b^2n-ca}(1+n)^2=0$

Step-by-step explanation:

Given quadratic equation

$ax^2+bx+c=0$

Let one root of the given quadratic equation be α

Then the other root will be = nα

Sum of the roots = -b/a

or $\alpha+n\alpha=-\frac{b}{a}$

or, $\alpha(1+n)=-\frac{b}{a}$   ............... (1)

Product of the roots = c/a

or, $\alpha \times n\alpha=\frac{c}{a}$

or, $n\alpha^2=\frac{c}{a}$

or, $\alpha^2=\frac{c}{na}$

Suqaring eq (1) and putting the value of α² in it

$\alpha^2(1+n)^2=(-\frac{b}{a})^2$

$\implies \frac{c}{na}(1+n)^2=\frac{b^2}{a^2}$

$\implies ca}(1+n)^2=b^2n$

$\implies b^2n-ca}(1+n)^2=0$

This is the required condition.

Hope it is helpful.

Answer 2

Refer The Attachment ⬆️

$\pink{\underline{\underline{\bf{Required\;AnSweR:}}}}$

The Condition is $\boxed{\red{\sf{nb² = (n+1)²ac}}}$

Hope It Helps You ✌️