To determine which one is a factor of the quadratic equation [tex]\(x^2 - 9x + 14\)[/tex], we should factorize the quadratic polynomial. Let's go through the steps of factorization carefully:
1. Given Polynomial:
[tex]\[
x^2 - 9x + 14
\][/tex]
2. Identify the Coefficients:
Here, the coefficient of [tex]\(x^2\)[/tex] is 1, the coefficient of [tex]\(x\)[/tex] is -9, and the constant term is 14.
3. Factorization Method:
We need to express the quadratic polynomial in the form:
[tex]\[
x^2 - 9x + 14 = (x - a)(x - b)
\][/tex]
where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are numbers that need to be determined.
4. Find [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\(a\)[/tex] and [tex]\(b\)[/tex] should satisfy two conditions:
- The product [tex]\(a \cdot b = 14\)[/tex]
- The sum [tex]\(a + b = 9\)[/tex]
5. Determine the Pair (a, b):
We need to find pairs of numbers that multiply to 14 and add up to 9.
[tex]\[
\begin{array}{ccc}
7 \cdot 2 &= 14 & \quad 7 + 2 = 9 \\
-7 \cdot (-2) &= 14 & \quad -7 + (-2) = -9 \quad (\text{wrong sum}) \\
-2 \cdot (-7) &= 14 & \quad -2 + (-7) = -9 \quad (\text{wrong sum}) \\
\end{array}
\][/tex]
The correct pair is [tex]\(7\)[/tex] and [tex]\(2\)[/tex].
6. Write the Factored Form:
Substituting [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into our factor form:
[tex]\[
x^2 - 9x + 14 = (x - 7)(x - 2)
\][/tex]
Therefore, the factors are [tex]\((x - 7)\)[/tex] and [tex]\((x - 2)\)[/tex].
7. Verify Factors:
- Option [tex]\(x - 9\)[/tex] does not satisfy the factorization.
- Option [tex]\(x - 2\)[/tex] is a factor.
- Option [tex]\(x + 5\)[/tex] does not match the factorization.
- Option [tex]\(x + 7\)[/tex] does not match the factorization (it should be [tex]\(x - 7\)[/tex]).
Conclusion:
The factor of [tex]\(x^2 - 9x + 14\)[/tex] from the given options is:
[tex]\[
x - 2
\][/tex]
