Sure! Let's go through the problem step by step:
### Factored Form
The equation [tex]\( a(x - p)(x - q) \)[/tex] is given in factored form. This form shows that the roots (or solutions) of the quadratic equation are [tex]\( x = p \)[/tex] and [tex]\( x = q \)[/tex]. The coefficient [tex]\( a \)[/tex] affects the width and direction of the parabola represented by the quadratic equation (whether it opens upwards or downwards if [tex]\( a \)[/tex] is positive or negative, respectively).
### Standard Form
To convert the factored form [tex]\( a(x - p)(x - q) \)[/tex] into the standard form [tex]\( ax^2 + bx + c \)[/tex], we need to expand the expression:
1. Distribute:
[tex]\[
(x - p)(x - q) = x^2 - (p + q)x + pq
\][/tex]
2. Multiply by [tex]\( a \)[/tex]:
[tex]\[
a(x^2 - (p + q)x + pq) = ax^2 - a(p + q)x + apq
\][/tex]
Therefore, the standard form of the quadratic equation [tex]\( a(x - p)(x - q) \)[/tex] is [tex]\( ax^2 - a(p + q)x + apq \)[/tex].
### Summary
Putting it all together, we have:
- Factored form: [tex]\( a(x - p)(x - q) \)[/tex]
- Standard form: [tex]\( ax^2 - a(p + q)x + apq \)[/tex]
But since the problem asks for a general representation:
- Factored form: [tex]\( a(x-p)(x-q) \)[/tex]
- Standard form: [tex]\( ax^2 + bx + c \)[/tex]
These two forms are equivalent representations of the same quadratic equation.
