The expression [tex]\((x-p)(x-q)\)[/tex] is a factored form of a quadratic equation. Let's delve into what this means and how it relates to finding the roots of the quadratic equation:
1. Understanding the Expression:
[tex]\((x - p)(x - q)\)[/tex] represents a quadratic expression where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are constants. This is the product of two linear expressions.
2. Setting the Expression to Zero:
To find the solutions or the roots of the equation, we set the expression equal to zero:
[tex]\[
(x - p)(x - q) = 0
\][/tex]
3. Zero Product Property:
The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. So, we can set each factor equal to zero separately:
[tex]\[
x - p = 0
\][/tex]
[tex]\[
x - q = 0
\][/tex]
4. Solving for [tex]\(x\)[/tex]:
Solving these equations gives us:
[tex]\[
x = p
\][/tex]
[tex]\[
x = q
\][/tex]
These values, [tex]\(x = p\)[/tex] and [tex]\(x = q\)[/tex], are the solutions or roots of the quadratic equation represented by [tex]\((x - p)(x - q) = 0\)[/tex].
5. Conclusion:
The process of expressing the quadratic equation in this form and finding its roots by setting each factor to zero is known as factoring. Factoring is a key method in algebra for solving quadratic equations, and it directly provides us with the roots of the equation.
Thus, the expression [tex]\((x-p)(x-q) \longrightarrow x = p\)[/tex] or [tex]\(x = q\)[/tex] illustrates the concept of factoring in solving quadratic equations.
