select one factor of x^2 +16x+63
factor always come in pairs
so
x^2 + 16x + 63
= (x+7)(x+9)
To determine a factor of the given quadratic equation, x^2 + 16x + 63, we need to identify a pair of numbers whose product is the constant term (63) and whose sum is the coefficient of the linear term (16).
In this case, the constant term is 63, and the coefficient of the linear term is 16. We need to find two numbers whose product is 63 and whose sum is 16.
To make this process systematic, we can find the prime factorization of 63: 63 = 3 * 3 * 7.
Now let's consider all possible pairs of these prime factors whose sum is 16:
1. 3 + 3 = 6
2. 3 + 7 = 10
3. 7 + 3 = 10
4. 3 * 3 = 9
5. 3 * 7 = 21
6. 7 * 3 = 21
None of these pairs adds up to 16, so there is no factor of x^2 + 16x + 63 where the coefficient of x is equal to 1. Therefore, x^2 + 16x + 63 does not have any linear factors (expressions of the form x + a or x - a).
However, it is important to note that x^2 + 16x + 63 can still be factored using methods such as factoring by grouping, completing the square, or using the quadratic formula. These methods would involve considering factors other than linear factors of the form x + a or x - a.