Certainly! Let's analyze the given expression [tex]\((x-p)(x-q)\)[/tex].
To solve for [tex]\(x\)[/tex], we need to understand the concept being illustrated here. This expression is a product of two binomials. For a product of two quantities to be zero, at least one of the quantities must be zero. Mathematically, this concept is captured by the Zero Product Property.
Step-by-Step Solution:
1. Consider the product [tex]\((x-p)(x-q) = 0\)[/tex].
2. According to the Zero Product Property, if the product of two terms is zero, then at least one of the terms must be zero.
3. This gives us two possible equations:
- [tex]\(x - p = 0\)[/tex]
- [tex]\(x - q = 0\)[/tex]
4. Solving these equations individually:
- From [tex]\(x - p = 0\)[/tex], we get [tex]\(x = p\)[/tex].
- From [tex]\(x - q = 0\)[/tex], we get [tex]\(x = q\)[/tex].
So, the solutions to the equation [tex]\((x-p)(x-q) = 0\)[/tex] are [tex]\(x = p\)[/tex] or [tex]\(x = q\)[/tex].
Therefore, [tex]\((x-p)(x-q) \rightarrow x=p \text{ or } x=q\)[/tex] illustrates the Zero Product Property.
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