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

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

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

Let's substitute these values into the quadratic formula:

\[
x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1}
\]

Simplifying further:

\[
x = \frac{-2 \pm \sqrt{4 + 24}}{2}
\]

\[
x = \frac{-2 \pm \sqrt{28}}{2}
\]

The square root of 28 can be simplified to \(2\sqrt{7}\):

\[
x = \frac{-2 \pm 2\sqrt{7}}{2}
\]

Divide all terms by 2:

\[
x = \frac{-2}{2} \pm \frac{2\sqrt{7}}{2}
\]

\[
x = -1 \pm \sqrt{7}
\]

Thus, the two solutions for the equation \(x^2 + 2x - 6 = 0\) are:

\[
x_1 = -1 + \sqrt{7}, \quad x_2 = -1 - \sqrt{7}
\]

Would you like more details or have any questions about this solution?

### Related Questions:
1. What are the steps to solve a quadratic equation by completing the square?
2. How do you find the vertex of a quadratic function?
3. How can the quadratic formula be derived from completing the square?
4. What does the discriminant tell us about the roots of a quadratic equation?
5. How can you solve a quadratic equation graphically?

### Tip:
Always check the discriminant (\(b^2 - 4ac\)) before solving a quadratic equation; it helps determine the nature and number of the roots.

Learn how to solve the quadratic equation x^2 + 2x - 6 = 0 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula. Suitable for students in Grades 9-12.

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