To find the factors of the expression 2x² + 3x + 54, we can take a systematic approach.
First, we need to check if this quadratic expression can be factored easily. A quadratic expression in the form of ax² + bx + c can be factored by looking for two numbers that multiply to ac (where c is the constant term) and add up to b (the coefficient of x).
In our case, we have:
- a = 2
- b = 3
- c = 54
Calculating ac gives us:
ac = 2 * 54 = 108
Now, we need to find two numbers that multiply to 108 and add up to 3.
The factors of 108 (such as 1, 2, 3, 6, 9, 12, 18, 27, 36, 54, and 108) do not yield any pair of numbers that satisfies the requirement of adding up to 3.
Since we cannot find two such integers, it indicates that this quadratic cannot be factored easily with integer coefficients. Therefore, we can conclude that:
The expression 2x² + 3x + 54 does not have factors in the set of real numbers, indicating it is a composite expression that cannot be factored further using simple integer factorization.