The given problem suggests that you are working with a quadratic equation of the form:

$x^2 - (S)x + P = 0$

where:

• $S$ is the sum of the roots, and
• $P$ is the product of the roots.

From the problem:

• The sum $S = 2$,
• The product $P = 8$.

Substituting these values into the general form:

$x^2 - (2)x + 8 = 0$

This is the quadratic equation you're looking for:

$x^2 - 2x + 8 = 0$

Now let's solve this equation:

1. The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by:

$\Delta = b^2 - 4ac$

For $x^2 - 2x + 8 = 0$:

• $a = 1$, $b = -2$, and $c = 8$,

$\Delta = (-2)^2 - 4(1)(8) = 4 - 32 = -28$

Since the discriminant is negative, the roots of the equation are complex numbers.

2. The roots of the quadratic equation are given by the formula:

$x = \frac{-b \pm \sqrt{\Delta}}{2a}$

Substituting the values:

$x = \frac{-(-2) \pm \sqrt{-28}}{2(1)} = \frac{2 \pm \sqrt{-28}}{2} = \frac{2 \pm 2i\sqrt{7}}{2}$

Simplifying:

$x = 1 \pm i\sqrt{7}$

Thus, the roots of the equation are:

$x = 1 + i\sqrt{7} \quad \text{and} \quad x = 1 - i\sqrt{7}$

Would you like further details or have any questions?

Here are some related questions for you to explore:

1. How do we verify if a quadratic equation has real or complex roots?
2. Can you derive the quadratic formula from the standard quadratic equation?
3. How does the discriminant determine the nature of the roots?
4. What are some examples of real-life applications of quadratic equations?
5. Can you explain how to factor a quadratic equation if the discriminant is zero?

Tip: When the discriminant is negative, the roots are always complex, involving an imaginary number.