The product of the roots of a quadratic equation is 
 
\[\dfrac{8}{9}\] and the sum of the roots is 
\[2\].
We are given that the equation is 
\[ax^2 -18x + c = 0\].
Find the values of the coefficients 
\[a\] and 
\[c\].

To find the values of the coefficients \(a\) and \(c\) in the quadratic equation \(ax^2 - 18x + c = 0\), we can use the relationships between the roots of a quadratic equation and its coefficients.

For a general quadratic equation \(ax^2 + bx + c = 0\), the sum and product of the roots are given by:

\[
\text{Sum of the roots } (\alpha + \beta) = -\frac{b}{a}
\]

\[
\text{Product of the roots } (\alpha \cdot \beta) = \frac{c}{a}
\]

In the given quadratic equation, \(b = -18\), and we are provided with:
- The sum of the roots \( \alpha + \beta = 2 \)
- The product of the roots \( \alpha \cdot \beta = \frac{8}{9} \)

### Step 1: Find \(a\)
Using the sum of the roots:

\[
\alpha + \beta = -\frac{-18}{a} = \frac{18}{a}
\]

We are given that \( \alpha + \beta = 2 \). So:

\[
\frac{18}{a} = 2
\]

To solve for \( a \), multiply both sides by \(a\) and then divide by 2:

\[
18 = 2a \implies a = \frac{18}{2} = 9
\]

Thus, \( a = 9 \).

### Step 2: Find \(c\)
Using the product of the roots:

\[
\alpha \cdot \beta = \frac{c}{a}
\]

Substituting the values \( \alpha \cdot \beta = \frac{8}{9} \) and \( a = 9 \):

\[
\frac{8}{9} = \frac{c}{9}
\]

To solve for \( c \), multiply both sides by 9:

\[
c = 8
\]

Thus, \( c = 8 \).

### Final Answer:
The values of the coefficients are \( \boxed{a = 9} \) and \( \boxed{c = 8} \).

Would you like more details or have any questions?

Here are five related questions to explore further:
1. How would the equation change if the sum of the roots was negative?
2. Can you determine the roots of the equation with these coefficients?
3. What is the discriminant of the quadratic equation with the found values of \(a\) and \(c\)?
4. How do the values of \(a\) and \(c\) affect the graph of the quadratic function?
5. How would the problem change if the product of the roots were \(\frac{8}{27}\) instead of \(\frac{8}{9}\)?

**Tip:** Always verify your solution by substituting the calculated values back into the original equation to ensure consistency with the given conditions.

Learn how to determine coefficients of a quadratic equation given the product and sum of its roots. This problem involves solving for coefficients using the relationships between roots and coefficients. Suitable for high school students studying algebra and quadratic equations.

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