Animated Solution for Mathematics - Quadratic Equations: Let $S=\{x\in \mathbb{R}:x\ge 0\text{ and }2|\sqrt{x}-3|+\sqrt{x}(\sqrt{x}-6)+6=0\}$. Then $S$ :

Imagine you are standing at the base of a mountain, looking up at a path that seems impossibly steep. That is how many students feel when they first encounter an equation like:

It looks like a chaotic mix of radicals and absolute values. But in mathematics, as in life, complexity is often just simplicity in disguise. Let us break this down together.

First, let us look at the term $\sqrt{x}(\sqrt{x}-6)+6$. If we expand this, we get $x-6\sqrt{x}+6$.

Now, look closely at the structure. Does it remind you of anything? It looks suspiciously like a quadratic expression.

If we recall the identity $(\sqrt{x}-3{)}^2=x-6\sqrt{x}+9$, we can see that our expression is almost a perfect square. By adding $9$ and subtracting $3$, we can rewrite it as:

This is the key that unlocks the entire problem.

Now, our equation becomes:

This looks much cleaner, doesn't it? We have the same term, $|\sqrt{x}-3|$, appearing in both parts.

Let us make our lives easier by using a substitution. Let $t=|\sqrt{x}-3|$. Since the modulus is always non-negative, we must remember the constraint $t\ge 0$.

Also, note that $t^2=(|\sqrt{x}-3|{)}^2=(\sqrt{x}-3{)}^2$. Now, the equation transforms into a simple quadratic:

Solving $t^2+2t-3=0$ is straightforward. We can factorize it as:

This gives us two potential values for $t$: $t=-3$ or $t=1$. But wait! We must return to our constraint.

We said $t\ge 0$. Therefore, we must reject $t=-3$. The only valid value is $t=1$.

Now we substitute back: $|\sqrt{x}-3|=1$. This means $\sqrt{x}-3=1$ or $\sqrt{x}-3=-1$.

Solving these two cases:

Case 1: $\sqrt{x}=4\Rightarrow x=16$.

Case 2: $\sqrt{x}=2\Rightarrow x=4$.

Both values are non-negative, satisfying our initial condition $x\ge 0$. Thus, the set $S=\{4,16\}$ contains exactly two elements.

We have climbed the mountain, and the view from the top is beautiful. Remember, the next time you see a terrifying equation, look for the hidden structure. It is almost always there, waiting for you to find it.