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.