Question

For which value of b can the expression x^(2) + bx + 18 be factored?

Answer 1

The expression x² + bx + 18 can be factored as (x + 3)(x + 6) for any value of b.

Step-by-step explanation:

This expression is a quadratic equation of the form ax² + bx + c = 0, where the constants are a = 1 and c = 18. To factor this quadratic equation, we need to find two numbers whose sum is equal to the coefficient of x (b) and whose product is equal to the constant term (c). In this case, we have a = 1, b = b, and c = 18.

So, we need to find two numbers whose sum is equal to b and whose product is equal to 18. The numbers that satisfy these conditions are 3 and 6, because 3 + 6 = 9 and 3 * 6 = 18.

Therefore, the expression x² + bx + 18 can be factored as (x + 3)(x + 6) for any value of b.

Answer 2

The expression x² + bx + 18 can be factored when b = -6 or b = 6.

Step-by-step explanation:

The expression x² + bx + 18 can be factored if the discriminant is a perfect square. The discriminant is given by the formula D = b² - 4ac, where a = 1, b is the coefficient of x, and c = 18.

Since we want D to be a perfect square, we need to find a value of b that makes b² - 4ac = a perfect square.

Let's substitute the values into the formula:

D = b² - 4ac = b² - 4(1)(18) = b² - 72

Now, we can simplify the expression b² - 72 to a perfect square:

b² - 72 = (b - 6)(b + 6)

Therefore, the expression x² + bx + 18 can be factored when b = -6 or b = 6.