You provided the quadratic equation $x^2 + 3x - 3 = 0$. To solve it, we can use the quadratic formula:

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

For the given equation $x^2 + 3x - 3 = 0$, the coefficients are:

• $a = 1$
• $b = 3$
• $c = -3$

Step-by-Step Solution:

1. Identify the discriminant: $\Delta = b^2 - 4ac = 3^2 - 4(1)(-3) = 9 + 12 = 21$

2. Substitute values into the quadratic formula: $x = \frac{-3 \pm \sqrt{21}}{2(1)} = \frac{-3 \pm \sqrt{21}}{2}$

3. Final answers: $x_1 = \frac{-3 + \sqrt{21}}{2}, \quad x_2 = \frac{-3 - \sqrt{21}}{2}$

These are the two solutions in their simplest form.

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:

1. How do you calculate the discriminant, and what does it tell you about the solutions?
2. What happens when the discriminant is negative, zero, or a perfect square?
3. How can you approximate the values of $x_1$ and $x_2$ numerically?
4. What are the implications of different values of $a$, $b$, and $c$ in a quadratic equation?
5. How would the solutions change if the quadratic equation had different coefficients?

Tip: Always check the discriminant before solving to know how many real or complex solutions to expect!