Solve the equation using the quadratic formula.
2 x squared minus 9 x minus 1 equals 0

The given quadratic equation is:

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

To solve this equation using the quadratic formula, we use the general formula:

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

Where the coefficients \( a \), \( b \), and \( c \) are taken from the standard quadratic equation \( ax^2 + bx + c = 0 \). In this case:

- \( a = 2 \)
- \( b = -9 \)
- \( c = -1 \)

### Step 1: Calculate the discriminant
The discriminant is:

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

Substitute the values of \( a \), \( b \), and \( c \):

\[
\Delta = (-9)^2 - 4(2)(-1) = 81 + 8 = 89
\]

### Step 2: Apply the quadratic formula
Now, plug the discriminant (\( \Delta \)) and the values of \( a \) and \( b \) into the quadratic formula:

\[
x = \frac{-(-9) \pm \sqrt{89}}{2(2)} = \frac{9 \pm \sqrt{89}}{4}
\]

### Step 3: Simplify the result
This gives two possible solutions for \( x \):

\[
x_1 = \frac{9 + \sqrt{89}}{4}, \quad x_2 = \frac{9 - \sqrt{89}}{4}
\]

These are the exact solutions. If you need approximate values, they would be:

\[
x_1 \approx \frac{9 + 9.434}{4} \approx \frac{18.434}{4} \approx 4.609
\]
\[
x_2 \approx \frac{9 - 9.434}{4} \approx \frac{-0.434}{4} \approx -0.109
\]

So, the approximate solutions are:

\[
x_1 \approx 4.609, \quad x_2 \approx -0.109
\]

Would you like further details or have any questions?

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Here are five related questions to expand your understanding:
1. How is the discriminant used to determine the nature of the roots of a quadratic equation?
2. Can you derive the quadratic formula from the process of completing the square?
3. How would the quadratic formula change if the equation had complex roots?
4. What are some practical applications of quadratic equations in physics or economics?
5. How would this solution change if the quadratic equation had been multiplied by a constant factor?

**Tip:** Always check the discriminant first when solving quadratic equations—it tells you if the roots are real or complex!

Solve the equation using the quadratic formula. 2x^2 - 9x - 1 = 0

Learn how to solve the quadratic equation 2x^2 - 9x - 1 using the quadratic formula. This solution breaks down the steps, calculates the discriminant, and provides both exact and approximate values for the roots. Suitable for students in Grades 8-10.

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