Final answer:
To factor the trinomial n² + 21n + 54, we find the factor pair of the constant term (+54) that adds up to the coefficient of the middle term (+21), which are 6 and 9, resulting in (n + 6)(n + 9). For quadratic equations that cannot be factored, the quadratic formula is used to determine the roots.
Step-by-step explanation:
To factor the trinomial n² + 21n + 54, we need to find two numbers that multiply to give +54 (the constant term) and add up to +21 (the coefficient of the middle term). By looking for factor pairs of 54, we find that 6 and 9 fulfill these conditions because 6 × 9 = 54 and 6 + 9 = 15. Therefore, the trinomial can be factored as (n + 6)(n + 9).
If we have a quadratic equation ax² + bx + c = 0, and factoring is not possible or straightforward, the quadratic formula can be used to find the roots of the equation. The quadratic formula is given by x = (-b ± √(b² - 4ac)) / (2a). This formula is essential for solving any quadratic equation when factoring is not an option.