Question

Let one root of the equation $x^2+\ell x+m=0$ is square of other root. If $m\in R$ then

• A. $\ell \in (-\infty ,\frac{1}{4}]\cup \left\{1\right\}$
• B. $\ell \in (-\infty ,o]$
• C. $\ell \in (-\infty ,\frac{1}{9}]$
• D. $\ell \in (\frac{1}{4},1]$

Answer 1

The correct option is A $\ell \in (-\infty ,\frac{1}{4}]\cup \left\{1\right\}$

Let a root of the quadratic equation $x^2+\ell x+m=0$ be $\alpha$.

$\Rightarrow$ another root is ${\alpha }^2$.

Now, $\alpha +{\alpha }^2=-l$-------(1)

$\alpha \cdot {\alpha }^2=m$

$\Rightarrow {\alpha }^3=m$

$\Rightarrow \alpha =m^{\frac{1}{3}}$

From (1),

$m^{\frac{2}{3}}+m^{\frac{1}{3}}=-l$

$m^{\frac{2}{3}}+m^{\frac{1}{3}}+l=0$

It is a quadratic equation $m^{\frac{1}{3}}$

Since, $m\in R$

So, $\Delta =(1{)}^2-4(1\left)\right(l)\ge 0$

$\Rightarrow 4l-1\ge 0$

$\Rightarrow l\le \frac{1}{4}$

Therefore, $l\in (-\infty ,\frac{1}{4})\cup \left\{1\right\}$