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To find the x-intercepts of the graph of the equation \( y = 2x^2 + x - 3 \), we set \( y = 0 \) and solve for \( x \).

1. **Set up the equation:**
   \[
   2x^2 + x - 3 = 0
   \]

2. **Use the quadratic formula:**
   The quadratic formula states that for any equation of the form \( ax^2 + bx + c = 0 \), the solutions for \( x \) are given by:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Here, \( a = 2 \), \( b = 1 \), and \( c = -3 \).

3. **Calculate the discriminant:**
   \[
   b^2 - 4ac = 1^2 - 4 \cdot 2 \cdot (-3) = 1 + 24 = 25
   \]

4. **Solve for \( x \):**
   Substitute into the quadratic formula:
   \[
   x = \frac{-1 \pm \sqrt{25}}{2 \cdot 2} = \frac{-1 \pm 5}{4}
   \]
   This gives two solutions:
   \[
   x = \frac{-1 + 5}{4} = \frac{4}{4} = 1
   \]
   and
   \[
   x = \frac{-1 - 5}{4} = \frac{-6}{4} = -\frac{3}{2}
   \]

5. **Conclusion:**
   The x-intercepts are \( x = -\frac{3}{2} \) and \( x = 1 \).

### Answer:
The correct choice is:
\[
\text{A) } -\frac{3}{2} \text{ and } 1
\]

Would you like a further breakdown or have any questions?

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### Related Questions to Explore:
1. How does the discriminant determine the nature of the roots of a quadratic equation?
2. How can the quadratic formula be derived from completing the square?
3. What are the applications of finding x-intercepts in real-world problems?
4. How can we graph the function \( y = 2x^2 + x - 3 \) to visualize the intercepts?
5. What is the relationship between the coefficients \( a \), \( b \), and \( c \) and the shape of the parabola?

### Tip:
Remember, the discriminant \( b^2 - 4ac \) tells you whether the quadratic equation has real or complex roots. A positive discriminant indicates two real roots, which correspond to x-intercepts on the graph.

What are the x-intercepts of the graph of the equation \( y = 2x^2 + x - 3 \)?

This guide explains how to find the x-intercepts of the quadratic equation \( y = 2x^2 + x - 3 \) using the quadratic formula. Learn how to solve for x in this algebra problem, suitable for students in Grades 8-10.

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