To solve the quadratic equation \( x^2 - 5x - 3 = 0 \), we can use the quadratic formula:

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

Here, the coefficients are:
- \( a = 1 \)
- \( b = -5 \)
- \( c = -3 \)

Now, let's plug in the values:

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-3)}}{2(1)}
\]

Simplifying further:

\[
x = \frac{5 \pm \sqrt{25 + 12}}{2}
\]

\[
x = \frac{5 \pm \sqrt{37}}{2}
\]

So the solutions are:

\[
x = \frac{5 + \sqrt{37}}{2} \quad \text{or} \quad x = \frac{5 - \sqrt{37}}{2}
\]

These are the exact solutions of the quadratic equation.

Would you like to see more details, or do you have any questions about this solution?

Here are 5 related questions to expand your understanding:
1. How do you complete the square to solve a quadratic equation?
2. What is the discriminant in a quadratic equation, and how does it affect the number of solutions?
3. How would the solutions change if the constant term in the quadratic equation were positive instead of negative?
4. Can you graph the equation \( x^2 - 5x - 3 = 0 \) and find the roots graphically?
5. How do you solve quadratic equations using factoring instead of the quadratic formula?

**Tip:** Always check if the discriminant (inside the square root) is positive, zero, or negative to determine the nature of the roots (real and distinct, real and equal, or complex).

Learn how to solve the quadratic equation x² - 5x - 3 = 0 step-by-step. Understand the quadratic formula and its application in finding the exact solutions. Suitable for advanced high school students and anyone studying algebra and quadratic equations.

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