Use the discriminant to determine whether the given equation has two irrational roots, two rational roots, one repeated root, or two complex roots. Do not solve. -3x^2 - 8x + 16 = 0

To determine the nature of the roots of the quadratic equation \(-3x^2 - 8x + 16 = 0\), we can use the **discriminant**. 

The discriminant \(\Delta\) for a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula:

\[
\Delta = b^2 - 4ac
\]

For the given equation \(-3x^2 - 8x + 16 = 0\), the coefficients are:
- \(a = -3\)
- \(b = -8\)
- \(c = 16\)

Now, we calculate the discriminant:

\[
\Delta = (-8)^2 - 4(-3)(16)
\]
\[
\Delta = 64 + 192
\]
\[
\Delta = 256
\]

### Interpretation of the Discriminant:
- If \(\Delta > 0\) and a perfect square, the equation has **two rational roots**.
- If \(\Delta > 0\) and not a perfect square, the equation has **two irrational roots**.
- If \(\Delta = 0\), the equation has **one repeated root**.
- If \(\Delta < 0\), the equation has **two complex roots**.

Since \(\Delta = 256\), and 256 is a perfect square, the equation has **two rational roots**.

Would you like further details or explanations on this? Here are five related questions:

1. How does the discriminant change for equations with different coefficients?
2. Can you always use the discriminant to determine the type of roots for any quadratic equation?
3. What are examples of equations with irrational roots, and how can we recognize them?
4. How do we find the actual roots of the equation after using the discriminant?
5. What happens if we change the sign of the leading coefficient in the quadratic equation?

**Tip:** A discriminant of zero always indicates that the quadratic equation has a repeated or "double" root.

Explore how to determine the nature of the roots for the quadratic equation -3x^2 - 8x + 16 using the discriminant. Learn key algebraic concepts such as the discriminant formula and its application. Suitable for students in Grades 9-11.

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