The ratio of the two roots of the quadratic equation $ax^2 + bx + c = 0$ is given as $4:5$.

Let's denote the roots by $r_1$ and $r_2$, such that: $r_1 : r_2 = 4 : 5$ This means that $r_1 = 4k$ and $r_2 = 5k$ for some constant $k$.

By Vieta's formulas for a quadratic equation $ax^2 + bx + c = 0$, we know:

1. The sum of the roots $r_1 + r_2 = -\frac{b}{a}$,
2. The product of the roots $r_1 \times r_2 = \frac{c}{a}$.

Step 1: Sum of the Roots

The sum of the roots is: $r_1 + r_2 = 4k + 5k = 9k$ According to Vieta's formula, this sum is equal to: $-\frac{b}{a} = 9k$

Step 2: Product of the Roots

The product of the roots is: $r_1 \times r_2 = 4k \times 5k = 20k^2$ According to Vieta's formula, this product is equal to: $\frac{c}{a} = 20k^2$

Step 3: Value of $ac$

From the product formula, we have: $\frac{c}{a} = 20k^2$ Thus: $c = 20k^2 a$ Hence, the value of $ac$ is: $ac = a \times 20k^2 = 20k^2 a$

Thus, the value of $ac$ is $20k^2 a^2$, which depends on the value of $k$ and $a$. However, in terms of the ratio, the product remains 20 times a square term.

Would you like to go further in this? Here are 5 questions that relate:

1. How do we find the roots explicitly using a quadratic formula?
2. Can the constant $k$ be determined from additional information?
3. How does the discriminant of the quadratic equation relate to the nature of the roots?
4. What happens if the ratio of roots changes?
5. How do you derive Vieta's formulas?

Tip: Understanding Vieta's relations helps in relating the sum and product of roots to coefficients of quadratic equations effectively.