The expression you provided, \(2x^2 + 3x - 2\), is a quadratic equation. To find its solutions (the values of \(x\) that satisfy this equation), we can either factor it, complete the square, or use the quadratic formula. 

Let's use the **quadratic formula**, which is:

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

For the equation \(2x^2 + 3x - 2 = 0\), the coefficients are:
- \(a = 2\)
- \(b = 3\)
- \(c = -2\)

Now, let's substitute these values into the quadratic formula.

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

\[
x = \frac{-3 \pm \sqrt{9 + 16}}{4}
\]

\[
x = \frac{-3 \pm \sqrt{25}}{4}
\]

\[
x = \frac{-3 \pm 5}{4}
\]

So, the two solutions are:

\[
x_1 = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}
\]

\[
x_2 = \frac{-3 - 5}{4} = \frac{-8}{4} = -2
\]

Thus, the solutions are \(x = \frac{1}{2}\) and \(x = -2\).

Would you like more details on any of the steps?

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Here are five related questions to expand on this topic:
1. How do you complete the square for this quadratic equation?
2. Can you graph the equation \(2x^2 + 3x - 2\) and interpret the roots visually?
3. What are the properties of the vertex for the quadratic equation \(2x^2 + 3x - 2\)?
4. How do you factor a quadratic equation when factoring is possible?
5. What are the differences between the quadratic formula and factoring methods?

**Tip**: Always check the discriminant, \(b^2 - 4ac\), before solving a quadratic equation. It tells you how many real solutions exist.

Learn how to solve the quadratic equation 2x^2 + 3x - 2 with a detailed step-by-step solution using the quadratic formula. This problem covers key algebraic concepts, including the quadratic formula and discriminant analysis. Suitable for students in Grades 8-10.

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