To solve the equation \( x^2 - 9x + 8 = 0 \) by factoring, we first need to express the quadratic in the form \( (x - p)(x - q) = 0 \), where \( p \) and \( q \) are the roots we need to find.
We can start by looking for two numbers that multiply to the constant term (8) and add up to the linear coefficient (-9).
The numbers that satisfy this condition are -1 and -8, since:
- \((-1) \times (-8) = 8\)
- \((-1) + (-8) = -9\)
Thus, we can factor the quadratic equation as follows:
\[ x^2 - 9x + 8 = (x - 1)(x - 8) \]
Setting this equal to zero gives us:
\[ (x - 1)(x - 8) = 0 \]
Now, we can set each factor equal to zero:
- \( x - 1 = 0 \) → \( x = 1 \)
- \( x - 8 = 0 \) → \( x = 8 \)
So, the solutions to the equation \( x^2 - 9x + 8 = 0 \) are:
\[ x = 1 \quad \text{and} \quad x = 8 \]