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: $\text{A) } -\frac{3}{2} \text{ and } 1$

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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.