To solve the quadratic equation $4x^2 - 15x - 503$, we will use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 4$, $b = -15$, and $c = -503$.

1. Calculate the discriminant ($b^2 - 4ac$): $(-15)^2 - 4 \cdot 4 \cdot (-503) = 225 + 8048 = 8273$

2. Substitute the values into the quadratic formula: $x = \frac{15 \pm \sqrt{8273}}{8}$

Therefore, the solutions are: $x = \frac{15 + \sqrt{8273}}{8}$ $x = \frac{15 - \sqrt{8273}}{8}$

Would you like more details or have any questions?

Here are some questions you might find useful:

1. How do you approximate the value of $\sqrt{8273}$?
2. Can you solve a quadratic equation by factoring instead of using the quadratic formula?
3. What are the steps to complete the square for this equation?
4. How can you determine if a quadratic equation has real or complex roots?
5. What are some methods to graph a quadratic equation?

Tip: Always simplify the discriminant first to determine the nature of the roots (real or complex) before proceeding with the quadratic formula.