Question

যদি
$ax {}^{2} + bx + c = 0$
দ্বিঘাত সমীকরণের বীজদ্বয়ের অনুপাত 1;r হয়, তবে,দেখাও যে
$\frac{(r + 1) {}^{2} }{r} = \frac{b {}^{2} }{ac}$
​

Answer 1

Step-by-step explanation:

Let the roots be $\alpha \:\: \& \:\: r\alpha$

Then,

$\alpha + r \alpha = - \frac{b}{a}$

And,

$r { \alpha }^{2} = \frac{c}{a}$

$\implies \: { \alpha }^{2} = \frac{c}{ar}$

And,

$\alpha (1 + r) = - \frac{b}{a} \\ \implies \: \alpha = - \frac{b}{a(r + 1)}$

So,

$\bigg( - \frac{b}{a(r + 1)} \bigg)^{2} = \frac{c}{ar}$

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

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

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