To solve the quadratic equation \( x^2 - 7x - 4 = 0 \), we can use the quadratic formula:

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

For the given equation \( x^2 - 7x - 4 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -7 \)
- \( c = -4 \)

Substitute these values into the quadratic formula:

\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-4)}}{2(1)}
\]

Simplifying further:

\[
x = \frac{7 \pm \sqrt{49 + 16}}{2}
\]
\[
x = \frac{7 \pm \sqrt{65}}{2}
\]

So the solutions are:

\[
x = \frac{7 + \sqrt{65}}{2} \quad \text{or} \quad x = \frac{7 - \sqrt{65}}{2}
\]

These are the exact solutions to the equation. If you want the decimal approximations:

\[
x \approx 7.53 \quad \text{or} \quad x \approx -0.53
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you derive the quadratic formula?
2. Can this quadratic equation be solved by completing the square?
3. How do you interpret the discriminant in the quadratic formula?
4. What are some real-life applications of solving quadratic equations?
5. How would the solutions change if the constant term was positive?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) first to determine the nature of the roots (real or complex).

Learn how to solve the quadratic equation x^2 - 7x - 4 = 0 with a detailed step-by-step solution using the quadratic formula. Understand the coefficients and discriminant, and find both exact and approximate solutions. Suitable for high school students in Grades 10-12.

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