Sure, let's analyze the given equation step-by-step and simplify it as required.
The equation provided is:
[tex]\[ z = px + qy + 2 \sqrt{pq} \][/tex]
### Step 1: Identifying the components
In this equation, we have:
- [tex]\( p \)[/tex] and [tex]\( q \)[/tex] which are coefficients.
- [tex]\( x \)[/tex] and [tex]\( y \)[/tex] which are variables.
- [tex]\( \sqrt{pq} \)[/tex] which signifies the square root of the product of [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
### Step 2: Simplification
Let's analyze why this form might be considered simplified.
#### Breaking it down:
1. Term 1: [tex]\( px \)[/tex] - This term represents the product of coefficient [tex]\( p \)[/tex] and variable [tex]\( x \)[/tex].
2. Term 2: [tex]\( qy \)[/tex] - Similarly, this term represents the product of coefficient [tex]\( q \)[/tex] and variable [tex]\( y \)[/tex].
3. Term 3: [tex]\( 2 \sqrt{pq} \)[/tex] - This term includes the square root, which needs to be combined with the factor 2.
### Combining Terms:
There is no further simplification achievable by combining terms due to the nature of these distinct parts. The equation as given:
[tex]\[ z = px + qy + 2 \sqrt{pq} \][/tex]
is in fact already in its simplest form. Each term is a fundamental component involving variables and coefficients and includes an additional constant [tex]\( 2 \sqrt{pq} \)[/tex] that cannot be further simplified or combined in any meaningful way.
### Conclusion:
Hence, the simplified form of the given equation remains:
[tex]\[ z = px + qy + 2 \sqrt{pq} \][/tex]
This final expression is the solution in its most simplified and clear form, representing the relationships between all the variables and coefficients in the original equation accurately.
