12. The equation a * x ^ 2 + bx + c = 0 where a, b, c are real numbers, has one root greater than 2 and the other root less than zero. Which of the following is necessarily true?

Given the quadratic equation \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are real numbers, we know the roots of the equation satisfy the following conditions:

- One root is greater than 2.
- The other root is less than zero.

Let the roots of the equation be \( r_1 \) and \( r_2 \). We know:

\[
r_1 > 2 \quad \text{and} \quad r_2 < 0.
\]

Using Vieta's formulas, which relate the coefficients of the quadratic equation to its roots, we have:

1. The sum of the roots: 

   \[
   r_1 + r_2 = -\frac{b}{a}.
   \]

2. The product of the roots:

   \[
   r_1 \cdot r_2 = \frac{c}{a}.
   \]

### Analysis of Each Condition:

Given that \( r_1 > 2 \) and \( r_2 < 0 \), let's determine which statements are necessarily true:

1. **Sum of the Roots:**
   
   Since \( r_1 \) is positive and larger than 2, and \( r_2 \) is negative, the sum \( r_1 + r_2 \) could be either positive, negative, or zero, depending on the magnitudes of \( r_1 \) and \( r_2 \). Therefore, \( -\frac{b}{a} \) is not necessarily positive or negative.

2. **Product of the Roots:**

   Since \( r_1 > 2 \) and \( r_2 < 0 \), their product \( r_1 \cdot r_2 \) must be negative. Therefore, \( \frac{c}{a} < 0 \).

**Conclusion:**

The statement that is necessarily true is:

\[
\boxed{\frac{c}{a} < 0}
\]

### Follow-up Questions:

1. What happens to the roots if \( b = 0 \) in the given quadratic equation?
2. How do the signs of \( a, b, \) and \( c \) affect the nature of the roots?
3. What does the discriminant \( b^2 - 4ac \) tell us about the roots of the equation?
4. How would the graph of the function \( y = ax^2 + bx + c \) look given the conditions on the roots?
5. What additional information can be deduced if \( r_1 \) and \( r_2 \) are integers?

**Tip:** When analyzing quadratic equations, always relate the properties of the coefficients to the nature of the roots using Vieta's formulas and the discriminant.

The equation a * x^2 + bx + c = 0 where a, b, c are real numbers, has one root greater than 2 and the other root less than zero. Which of the following is necessarily true?

Explore the quadratic equation ax^2 + bx + c = 0, where one root is greater than 2 and the other is less than 0. This detailed solution uses Vieta's formulas to explain why the product of the roots must be negative. Ideal for students in Grades 10-12.

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