Let's solve the equation step-by-step:

The equation you gave is:

$2k^2 + 3k - 1 = 0$

This is a quadratic equation of the form $ax^2 + bx + c = 0$, where $a = 2$, $b = 3$, and $c = -1$.

Step 1: Use the quadratic formula

The quadratic formula is:

$k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute the values of $a$, $b$, and $c$:

$k = \frac{-3 \pm \sqrt{3^2 - 4(2)(-1)}}{2(2)}$

$k = \frac{-3 \pm \sqrt{9 + 8}}{4}$

$k = \frac{-3 \pm \sqrt{17}}{4}$

Step 2: Simplify

The solutions for $k$ are:

$k = \frac{-3 + \sqrt{17}}{4} \quad \text{or} \quad k = \frac{-3 - \sqrt{17}}{4}$

Final Answer:

The two solutions for $k$ are:

$k = \frac{-3 + \sqrt{17}}{4} \quad \text{and} \quad k = \frac{-3 - \sqrt{17}}{4}$

Would you like me to simplify further or provide more details on how to approach solving quadratic equations?

Here are some related questions:

1. What is the discriminant in the quadratic formula, and how does it affect the number of solutions?
2. Can you solve a quadratic equation by factoring? When is factoring possible?
3. How do you interpret the solutions of a quadratic equation geometrically on a parabola?
4. What is the significance of the term "vertex" in the graph of a quadratic equation?
5. How would you solve a quadratic equation using completing the square?

Tip: If the discriminant (the part inside the square root in the quadratic formula) is positive, there are two real solutions. If it's zero, there's exactly one real solution, and if it's negative, the solutions are complex numbers.