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:
- 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 \( c = 11 \):
- Calculate the discriminant: \[ b^2 - 4ac = 8^2 - 4 \cdot 1 \cdot 11 = 64 - 44 = 20 \]
- 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: \[ x = -4 + \sqrt{5} \quad \text{and} \quad x = -4 - \sqrt{5} \]