The equation $P(Q(x))=0$ means that the function $P$ composed with the function $Q$ results in a value of 0. In other words, the input $x$ is being transformed by $Q$, and then the output of $Q$ is being plugged into $P$, resulting in a final output of 0.
To solve $P(Q(x))=0$ for real roots, you need to first determine the function $Q(x)$ and then plug it into the function $P(x)$. This will result in a new equation, which can then be solved using various methods such as factoring, the quadratic formula, or graphing.
Specifying "real roots" is important because it indicates that we are only interested in solutions that are real numbers. In other words, we are looking for values of $x$ that will make the equation $P(Q(x))=0$ true in the real number system. This is important because some equations may have complex solutions, which are not considered real roots.
Yes, it is possible for $P(Q(x))=0$ to have more than one real root. This will depend on the specific functions $P(x)$ and $Q(x)$ and how they interact with each other. For example, if $P(x)$ is a quadratic function and $Q(x)$ is a linear function, the resulting equation $P(Q(x))=0$ could have two real roots.
The concept of solving $P(Q(x))=0$ for real roots has many real-life applications, such as in physics, engineering, and economics. For example, in physics, this equation can be used to model the motion of a projectile, where $P(x)$ represents the vertical position and $Q(x)$ represents the horizontal position. In engineering, this equation can be used to design circuits, where $P(x)$ represents the voltage and $Q(x)$ represents the current. In economics, this equation can be used to analyze supply and demand, where $P(x)$ represents the price and $Q(x)$ represents the quantity.