To find the roots of the quadratic function $p(x) = x^2 - 4x + 22$, we can apply the quadratic formula:

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

For the equation $p(x) = ax^2 + bx + c$, the coefficients are:

• $a = 1$
• $b = -4$
• $c = 22$

Substitute these into the quadratic formula:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(22)}}{2(1)}$ $x = \frac{4 \pm \sqrt{16 - 88}}{2}$ $x = \frac{4 \pm \sqrt{-72}}{2}$ $x = \frac{4 \pm \sqrt{72}i}{2}$ $x = \frac{4 \pm 6i}{2}$ $x = 2 \pm 3i$

Thus, the roots are: $x = 2 + 3i \quad \text{and} \quad x = 2 - 3i$

These are complex roots because the discriminant $b^2 - 4ac$ is negative.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you interpret complex roots geometrically on the complex plane?
2. Can all quadratic functions have complex roots?
3. What is the role of the discriminant in determining the nature of the roots?
4. How can you express a quadratic function in vertex form?
5. How would the graph of $p(x) = x^2 - 4x + 22$ look?

Tip: The discriminant helps determine if the roots are real or complex. A negative discriminant indicates complex roots.