To solve the quadratic equation \(x^2 + 3x - 6 = 0\), we can use the quadratic formula:

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

For the equation \(x^2 + 3x - 6 = 0\), the coefficients are:
- \(a = 1\)
- \(b = 3\)
- \(c = -6\)

### Step 1: Calculate the discriminant
The discriminant \(\Delta\) is given by:

\[
\Delta = b^2 - 4ac = (3)^2 - 4(1)(-6) = 9 + 24 = 33
\]

### Step 2: Apply the quadratic formula
Now, substitute the values of \(a\), \(b\), and \(\Delta\) into the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{33}}{2(1)} = \frac{-3 \pm \sqrt{33}}{2}
\]

### Final Answer:
Thus, the two solutions for \(x\) are:

\[
x = \frac{-3 + \sqrt{33}}{2} \quad \text{or} \quad x = \frac{-3 - \sqrt{33}}{2}
\]

These are the exact solutions. If you need the approximate decimal values:

\[
x_1 \approx 1.37 \quad \text{and} \quad x_2 \approx -4.37
\]

Would you like further details or have any questions?

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Here are five related questions to consider:
1. How does the discriminant affect the nature of the solutions?
2. Can you graph the function \(y = x^2 + 3x - 6\) and find the roots?
3. What happens if the discriminant is negative or zero?
4. How can you factorize a quadratic equation if possible?
5. Can you solve quadratic equations using a method other than the quadratic formula?

**Tip:** Always check the discriminant (\(\Delta\)) first, as it tells you whether the roots are real or complex.

Learn how to solve the quadratic equation x^2 + 3x - 6 = 0 with a detailed step-by-step solution. Understand the quadratic formula and its application. Suitable for high school students.

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