We are going to factor the quadratic expression x squared minus 9x plus 14. Factoring means finding two expressions that multiply to give the original. Our expression is in the general form a x squared plus b x plus c, where a equals 1, b equals negative 9, and c equals 14.
For a quadratic in the form x squared plus b x plus c, we need to find two numbers that multiply to give c and add to give b. In our case, c is 14 and b is negative 9. Let's list all factor pairs of 14 and check their sums.
Now we check each pair. One times fourteen equals fourteen, but one plus fourteen equals fifteen, not negative nine. Two times seven equals fourteen, but two plus seven equals nine, not negative nine. Negative one times negative fourteen equals fourteen, but negative one plus negative fourteen equals negative fifteen, not negative nine. Finally, negative two times negative seven equals fourteen, and negative two plus negative seven equals negative nine. This is our pair!
Now that we've identified negative two and negative seven as our factor pair, we can write the factored form. Since both numbers are negative, our factors are (x minus 2) and (x minus 7). So x squared minus 9x plus 14 factors to (x minus 2) times (x minus 7).
To verify our factoring is correct, let's expand (x minus 2) times (x minus 7) using the FOIL method. First terms: x times x equals x squared. Outer terms: x times negative 7 equals negative 7x. Inner terms: negative 2 times x equals negative 2x. Last terms: negative 2 times negative 7 equals 14. Combining these gives x squared minus 7x minus 2x plus 14. Simplifying like terms gives x squared minus 9x plus 14, which matches our original expression.
Let's summarize the complete solution. We started with x squared minus 9x plus 14. We needed to find two numbers that multiply to 14 and add to negative 9. We identified the pair negative 2 and negative 7. Therefore, the factored form is (x minus 2) times (x minus 7). This technique works for any quadratic expression where the coefficient of x squared is 1.