Animated Solution for Mathematics - Quadratic Equations: The sum of the solutions of the equation $|\sqrt{x}-2|+\sqrt{x}(\sqrt{x}-4)+2=0,(x>0)$ is equal to :

The given equation is:

At first glance, it looks like a chaotic mess of radicals and absolute values. However, in the world of JEE Advanced, chaos is just a hidden order waiting to be discovered.

Notice how $\sqrt{x}$ appears twice. Whenever you see a repeating structure, your first instinct should be to simplify the landscape. Let us introduce a new variable, $t=\sqrt{x}$.

Since the problem specifies $x>0$, we know that $t$ must also be positive. This simple substitution transforms our intimidating equation into:

Expanding the term $t(t-4)$ gives us $t^2-4t$. Now we have:

Look closely at the $t^2-4t$ part. It is almost a perfect square. If we add and subtract $4$, we get $(t^2-4t+4)-4$, which is $(t-2{)}^2-4$.

Substituting this back, the equation becomes:

This is the moment where the problem shifts from a struggle to a dance. We have successfully aligned the terms to match the modulus.

We know that for any real number $a$, $a^2=|a{|}^2$. This is a powerful tool in your JEE toolkit.

Let $u=|t-2|$. Since $u$ is a modulus, we must remember the constraint $u\ge 0$. Substituting $u$ into our equation, we get:

Factoring this is straightforward:

This gives us two potential values for $u$: $1$ and $-2$. But wait! Recall our constraint $u\ge 0$. We must reject $u=-2$ because a modulus cannot be negative. Thus, we are left with $u=1$.

Now, we simply retrace our steps. Since $u=|t-2|=1$, we have two cases:

This gives us $t=3$ and $t=1$. Finally, we return to our original variable $x$. Since $t=\sqrt{x}$, we have $x=t^2$.

Squaring our values for $t$, we get:

Both values are positive, satisfying our initial condition. The sum of these solutions is:

And there you have it—a complex problem dismantled by the power of substitution and logical consistency. Keep practicing this mindset, and you will find that even the toughest problems have a beautiful, logical soul.