To solve the quadratic equation $x^2 - 7x + 8 = 0$, we can use the quadratic formula:

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

where $a = 1$, $b = -7$, and $c = 8$.

Step 1: Substitute the values into the formula

1. Calculate $b^2 - 4ac$: $(-7)^2 - 4 \cdot 1 \cdot 8 = 49 - 32 = 17$

2. Write the expression for $x$: $x = \frac{7 \pm \sqrt{17}}{2}$

Step 2: Determine the solution set

The solutions are: $x = \frac{7 - \sqrt{17}}{2} \quad \text{and} \quad x = \frac{7 + \sqrt{17}}{2}$

Step 3: Match with the answer choices

Looking at the choices:

• Option (e): $\left\{ \frac{7 - \sqrt{17}}{2}, \frac{7 + \sqrt{17}}{2} \right\}$ matches the solution we found.

Answer:

The correct choice is e.

Would you like to see further details on this method?

Related Questions:

1. What is the quadratic formula and when is it used?
2. How do we determine if a quadratic equation has real solutions?
3. Can the quadratic formula be used for complex solutions?
4. What other methods can solve a quadratic equation besides the formula?
5. How does completing the square compare to using the quadratic formula?

Tip:

Always check the discriminant $b^2 - 4ac$ before using the quadratic formula; it reveals the nature of the roots.