Solve $$ x^{2}-9 x-52=0 $$ $$ x=\{\square\} $$

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To solve the quadratic equation $$ x^{2} - 9x - 52 = 0 $$, we can use the quadratic formula:

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

Where $$ a = 1 $$, $$ b = -9 $$, and $$ c = -52 $$.

1. **Calculate the discriminant ($$ \Delta $$)**:
   
   $$
   \Delta = b^2 - 4ac = (-9)^2 - 4(1)(-52) = 81 + 208 = 289
   $$

2. **Compute the square root of the discriminant**:
   
   $$
   \sqrt{289} = 17
   $$

3. **Find the two possible values of $$ x $$**:

$$
   x = \frac{9 \pm 17}{2}
   $$
   
   - **First solution**:
     
     $$
     x = \frac{9 + 17}{2} = \frac{26}{2} = 13
     $$
   
   - **Second solution**:
     
     $$
     x = \frac{9 - 17}{2} = \frac{-8}{2} = -4
     $$

4. **Write the solution set**:

$$
   x = \{13, -4\}
   $$

**Final Answer:**
$$
x = \{13, -4\}
$$
$$ x = 13 $$ or $$ x = -4 $$

Solve \( x^{2}-9 x-52=0 \) \( x=\{\square\} \)

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