Question

x^{2}+8x+11

Answer 1

The expression you provided, \( x^2 + 8x + 11 \), is a quadratic equation in standard form, where \( a = 1 \), \( b = 8 \), and \( c = 11 \).

### To factor the quadratic:
1. **Find two numbers that multiply to \( ac = 1 \cdot 11 = 11 \) and add to \( b = 8 \)**.
- The numbers 1 and 11 multiply to 11 but add up to 12, which does not fit.
- The numbers 2 and 5 do not fit either.

This means the quadratic does not factor nicely over the integers.

### To find the roots using the quadratic formula:
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Substituting \( a = 1 \), \( b = 8 \), and \( c = 11 \):
1. Calculate the discriminant:
\[
b^2 - 4ac = 8^2 - 4 \cdot 1 \cdot 11 = 64 - 44 = 20
\]
2. Use the quadratic formula:
\[
x = \frac{-8 \pm \sqrt{20}}{2 \cdot 1}
\]
\[
x = \frac{-8 \pm 2\sqrt{5}}{2}
\]
\[
x = -4 \pm \sqrt{5}
\]

### Conclusion:
The roots of the quadratic \( x^2 + 8x + 11 \) are:
\[
x = -4 + \sqrt{5} \quad \text{and} \quad x = -4 - \sqrt{5}
\]