https://file.aiquickdraw.com/mathbot/file/1728353056563fmjxxdha.jpg

The quadratic equation provided in the image is:

\[
4x^2 - 9x - 1 = 0
\]

We can solve this using the **quadratic formula**:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the given equation \(4x^2 - 9x - 1 = 0\), we have:
- \(a = 4\)
- \(b = -9\)
- \(c = -1\)

### Step-by-step solution:
1. **Calculate the discriminant**:
   \[
   \Delta = b^2 - 4ac = (-9)^2 - 4(4)(-1) = 81 + 16 = 97
   \]

2. **Substitute values into the quadratic formula**:
   \[
   x = \frac{-(-9) \pm \sqrt{97}}{2(4)} = \frac{9 \pm \sqrt{97}}{8}
   \]

So, the two solutions are:
\[
x_1 = \frac{9 + \sqrt{97}}{8}, \quad x_2 = \frac{9 - \sqrt{97}}{8}
\]

These are the exact forms of the solutions. 

If you want numerical approximations:
\[
x_1 \approx 2.36, \quad x_2 \approx -0.11
\]

Would you like further details on any of the steps?

### Related Questions:
1. What is the discriminant of a quadratic equation, and how does it affect the nature of the solutions?
2. How do you solve quadratic equations when the discriminant is negative?
3. Can you use factoring to solve \(4x^2 - 9x - 1 = 0\), or is the quadratic formula the best method here?
4. What are some real-life applications of quadratic equations?
5. How does the quadratic formula relate to completing the square?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) first to know if the solutions will be real or complex.

Use the quadratic formula to solve for x: 4x^2 - 9x - 1 = 0.

Learn how to solve the quadratic equation 4x^2 - 9x - 1 using the quadratic formula. This step-by-step guide covers the discriminant calculation and substitution into the formula, providing both exact and approximate solutions. Suitable for students in Grades 9-12.

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