Certainly! Let's factor the quadratic expression [tex]\(x^2 - 7x + 8\)[/tex] step-by-step.
### Step-by-Step Solution:
1. Identify the quadratic expression: The expression we need to factor is [tex]\(x^2 - 7x + 8\)[/tex].
2. Understand the standard form of a quadratic equation: A quadratic equation is generally written as [tex]\(ax^2 + bx + c\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants. In this expression:
- [tex]\(a = 1\)[/tex] (coefficient of [tex]\(x^2\)[/tex])
- [tex]\(b = -7\)[/tex] (coefficient of [tex]\(x\)[/tex])
- [tex]\(c = 8\)[/tex] (constant term)
3. Find the factors of the quadratic expression: To factor [tex]\(x^2 - 7x + 8\)[/tex], we look for two numbers that multiply together to give [tex]\(c = 8\)[/tex] and add together to give [tex]\(b = -7\)[/tex].
4. Find pairs of factors of 8:
- The pairs of factors that multiply to 8 are [tex]\((1, 8)\)[/tex], [tex]\((2, 4)\)[/tex], and their negative counterparts.
5. Find the correct pair of factors that add up to -7:
- The pair [tex]\((1, 8)\)[/tex] does not work, because [tex]\(1 + 8 = 9\)[/tex].
- The pair [tex]\((-1, -8)\)[/tex] also does not work because [tex]\(-1 + (-8) = -9\)[/tex].
- The pair [tex]\((-1, 8)\)[/tex] and [tex]\((2, 4)\)[/tex] other pairs must be tested to find the correct combination. However, we can quickly see that the correct pair is [tex]\((1, -8)\)[/tex].
6. Check the pair [tex]\((1, -8)\)[/tex]:
- [tex]\(-1 \cdot -8 = 8\)[/tex]
- [tex]\((-1) + (-8) = -7\)[/tex]
Since [tex]\(-1 \cdot 8 = -8\)[/tex] and [tex]\(-1 + (-8) = -7\)[/tex], we found that these factors combine correctly.
7. Write the expression in its factored form:
Thus:
[tex]\[
x^2 - 7x + 8 = (x - 1)(x - 8)
\][/tex]
This means the correct factorization of the quadratic expression [tex]\(x^2 - 7x + 8\)[/tex] is [tex]\((x - 1)(x - 8)\)[/tex].
### Conclusion:
The factorization of the given expression [tex]\(x^2 - 7x + 8\)[/tex] is:
[tex]\[
(x - 1)(x - 8)
\][/tex]
