Find the nature of the roots of 6x2 + 5x - 4 = 0.
Real and equal
Complex
Rational
Irrational

To determine the nature of the roots of the quadratic equation \(6x^2 + 5x - 4 = 0\), we use the discriminant formula:

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

For the equation \(ax^2 + bx + c = 0\), we have:
- \(a = 6\)
- \(b = 5\)
- \(c = -4\)

Now, calculate the discriminant:

\[
\Delta = 5^2 - 4(6)(-4) = 25 + 96 = 121
\]

Since the discriminant (\(\Delta = 121\)) is a positive perfect square, the roots of the equation are **real and rational**.

### Answer: Rational

Would you like further details or explanations on any step?

Here are 5 related questions for further exploration:
1. What happens to the nature of the roots if the discriminant is negative?
2. How would you solve \(6x^2 + 5x - 4 = 0\) to find the exact roots?
3. Can a quadratic equation have irrational roots? Under what conditions?
4. How can the quadratic formula be used to solve any quadratic equation?
5. What is the significance of a discriminant being zero?

**Tip:** If the discriminant is a perfect square, the roots are rational.

Find the nature of the roots of 6x^2 + 5x - 4 = 0.

Determine the nature of the roots for the quadratic equation 6x^2 + 5x - 4 = 0 using the discriminant formula. This problem involves key algebraic concepts, including the discriminant and the quadratic formula, making it suitable for students in Grades 8-10.

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