1 Use the quadratic formula to find the real solutions, if any, of the equation. $$ x^{2}+2 x-6=0 $$ Select the correct choice below and fill in any answer boxes within your choice. A. The solution set is $$ \{\square $$. (Simplify your answer. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. There are no real solutions.

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To solve the quadratic equation using the **quadratic formula**, we'll follow these steps:

**Given Equation:**
$$ x^{2} + 2x - 6 = 0 $$

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

**Identify the coefficients:**
- $$ a = 1 $$
- $$ b = 2 $$
- $$ c = -6 $$

**Calculate the discriminant ($$ \Delta $$):**
$$ \Delta = b^{2} - 4ac = (2)^{2} - 4(1)(-6) = 4 + 24 = 28 $$

Since the discriminant is positive ($$ \Delta = 28 > 0 $$), there are **two real solutions**.

**Apply the quadratic formula:**
$$ x = \frac{ -2 \pm \sqrt{28} }{2} $$

**Simplify the square root:**
$$ \sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7} $$

**Substitute back into the formula:**
$$ x = \frac{ -2 \pm 2\sqrt{7} }{2} $$
$$ x = -1 \pm \sqrt{7} $$

**Solution Set:**
$$ \left\{ -1 - \sqrt{7},\ -1 + \sqrt{7} \right\} $$

**Conclusion:**

**A.** The solution set is $$ \{\, -1 - \sqrt{7},\ -1 + \sqrt{7} \} $$.
The solution set is $$ \{-1 - \sqrt{7},\ -1 + \sqrt{7}\} $$.

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