Question

Let $\alpha$ and $\beta$ be the roots of the equation $x^2-(1-2a^2)x+(1-2a^2)=0$.Under what condition does the given equation have real roots.

• A. $a^2<\frac{1}{2}$
• B. $a^2>\frac{1}{2}$
• C. $a^2\le \frac{1}{2}$
• D. $a^2\ge \frac{1}{2}$

Answer 1

The correct option is D $a^2\ge \frac{1}{2}$

The given equation is $x^2-(1-2a^2)x+(1-2a^2)=0$

Now, roots will be real if $D\ge 0$.So

$D=b^2-4ac$ for equation $a{x}^2+bx+c=0$

Here,$a=1,b=-(1-2a^2),c=1-2a^2$

$D\ge 0$

$\Rightarrow (-(1-2a^2){)}^2-\mathrm{4.1.}(1-2a^2)\ge 0$

$\Rightarrow 1+4a^4-4a^2-4+8a^2\ge 0$

$\Rightarrow 4a^4+4a^2-3\ge 0$

$\Rightarrow 4a^4+6a^2-2a^2-3\ge 0$

$\Rightarrow 2a^2(2a^2+3)-1(2a^2+3)\ge 0$

$\Rightarrow (2a^2-1)(2a^2+3)\ge 0$

$\Rightarrow 2a^2+3$ will always be positive as $a^2\ge 0$

So $2a^2-1\ge 0$ $\Rightarrow a^2\ge \frac{1}{2}$

Hence, $D$ is correct.