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The task asks to find the intersections of the quadratic polynomial \( P_2(x) = a_2 x^2 + a_1 x + a_0 \), where the coefficients are:

\[
a_0 = -3, \quad a_1 = -2, \quad a_2 = 1
\]

Thus, the polynomial is:

\[
P_2(x) = x^2 - 2x - 3
\]

### 1. Intersection with the X-axis:
To find the intersection with the X-axis, we solve for \( P_2(x) = 0 \):

\[
x^2 - 2x - 3 = 0
\]

This is a quadratic equation, and we can solve it using the quadratic formula:

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

Here, \(a = 1\), \(b = -2\), and \(c = -3\). Substituting these values into the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)}
\]
\[
x = \frac{2 \pm \sqrt{4 + 12}}{2}
\]
\[
x = \frac{2 \pm \sqrt{16}}{2}
\]
\[
x = \frac{2 \pm 4}{2}
\]

Thus, the two solutions are:

\[
x = \frac{2 + 4}{2} = 3 \quad \text{and} \quad x = \frac{2 - 4}{2} = -1
\]

So, the intersections with the X-axis are at \(x = 3\) and \(x = -1\).

### 2. Intersection with the Y-axis:
The intersection with the Y-axis occurs when \(x = 0\):

\[
P_2(0) = (0)^2 - 2(0) - 3 = -3
\]

Thus, the intersection with the Y-axis is at \(y = -3\).

### Final Result:
- The intersections with the X-axis are \(x = 3\) and \(x = -1\).
- The intersection with the Y-axis is at \(y = -3\).

These results match the graph shown in **Figure 1**, confirming that the calculations are correct.

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Do you want more details or have any questions about this solution?

Here are some related questions to deepen understanding:

1. How does the quadratic formula work for any general quadratic equation?
2. What are the implications of the discriminant \(b^2 - 4ac\) in the quadratic formula?
3. How do we interpret the vertex of a quadratic function like \(P_2(x)\)?
4. What does it mean graphically when a quadratic function has no real roots?
5. How do we derive the standard form of a quadratic equation?

**Tip:** Always check if you can factor a quadratic equation before applying the quadratic formula. It can save time!

Para el polinomio de orden 2 de coeficientes dados en (2), obtengan analíticamente los valores de las intersecciones de la función P_2(x) con los ejes coordenados, comprobando que son los dadas en la Figura 1.

Find the intersections of the quadratic polynomial P_2(x) = x^2 - 2x - 3 with the X and Y axes using the quadratic formula. This detailed solution walks through the process of solving quadratic equations and confirming the graphical results.

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