Animated Solution for Mathematics - Functions: Let $[t]$ denote the greatest integer $\le t$. Then the equation in $x$, $[x{]}^2+2[x+2]-7=0$ has :

A Journey into $[x]$

Welcome, future engineers! Today, we are going to demystify the Greatest Integer Function, often called the floor function, denoted by $[x]$. It is a function that can look like a brick wall, but once you understand its internal logic, it becomes a playground for algebra.

Let us dive into the equation:

Our mission is to find how many values of $x$ satisfy this. Do not let the brackets intimidate you; we will break this down into a beautiful, logical sequence.

The Power of Integer Extraction

Before we start solving, we need to simplify the expression. The term $[x+2]$ is the first hurdle. Many students instinctively want to distribute the bracket, but that is a trap!

Instead, we use the fundamental property of the Greatest Integer Function: for any integer $n$, $[x+n]=[x]+n$. Since $2$ is clearly an integer, we can pull it out of the bracket.

This transforms our equation into:

Notice how the equation is already starting to look more manageable. By applying this property, we have removed the complexity of the addition inside the function.

The Quadratic Transformation

Now, let us expand the terms. We have $[x{]}^2+2[x]+4-7=0$. Combining the constants, $4-7$ gives us $-3$.

So, our equation simplifies to:

This is the "aha!" moment. If you look closely, this is not just any equation; it is a quadratic equation in disguise. Let us introduce a dummy variable, $y=[x]$.

Now, the equation becomes:

This is a standard quadratic that we can solve by factoring. We need two numbers that multiply to $-3$ and add to $2$. Those numbers are $3$ and $-1$.

Thus, we can write the equation as:

This gives us two possible values for $y$: $y=-3$ and $y=1$.

The Interval Trap

We have found the values for $y$, but remember, $y$ was just a placeholder for $[x]$. We must revert to our original variable.

We have two cases: $[x]=-3$ and $[x]=1$. This is where many students stop, thinking they have found two solutions. But wait!

The Greatest Integer Function $[x]=k$ does not mean $x=k$. It means $x$ is any number in the interval $[k,k+1)$.

For the first case, $[x]=-3$, this implies $x$ must be in the interval $[-3,-2)$. For the second case, $[x]=1$, this implies $x$ must be in the interval $[1,2)$.

The Infinite Reality

Look at these intervals: $[-3,-2)$ and $[1,2)$. These are not just single points; they are continuous segments on the real number line.

Each segment contains an infinite number of real values. Therefore, the original equation does not have two solutions, or four solutions; it has infinitely many solutions.

This problem is a classic JEE favorite because it tests whether you truly understand the definition of the function or if you are just blindly manipulating symbols. You have successfully navigated the trap, simplified the algebra, and uncovered the infinite nature of the solution set. Keep this logic in your toolkit, and no function will ever be able to stop you!