factoring ax^2+bx+c

30x^-24x-54

To factor the quadratic expression ax^2 + bx + c, where a, b, and c are constants, you can follow these steps:

Step 1: Ensure that the expression is in the form ax^2 + bx + c.
In your case, the quadratic expression is provided as 30x^2 - 24x - 54, which is already in the correct form.

Step 2: Check if the quadratic expression can be factored using the AC method.
To determine if the expression can be factored using the AC method, we need to find two numbers, let's call them p and q, whose sum is equal to the coefficient of the middle term (b) and whose product is equal to the multiplication of the coefficient of the first term (a) and the constant term (c).

In this case, a = 30, b = -24, and c = -54.

We need to find two numbers, p and q, such that p + q = -24 and p * q = (30) * (-54) = -1620.

To find p and q, we can look for pairs of numbers that multiply to -1620 and add up to -24. After checking different pairs of numbers, we find that p = -54 and q = 30 meet our requirements since -54 + 30 = -24 and -54 * 30 = -1620.

Step 3: Rewrite the middle term (bx) using p and q.
We rewrite the middle term (-24x) by splitting it into two terms using p and q:

30x^2 - 54x + 30x - 54
(30x^2 - 54x) + (30x - 54)

Step 4: Factor by grouping.
Now, we can factor by grouping. We take out the greatest common factor from each pair of terms:

6x(5x - 9) + 6(5x - 9)

Step 5: Factor out the common binomial.
We find that (5x - 9) is a common binomial, so we factor it out:

(5x - 9)(6x + 6)

Therefore, the factored form of the quadratic expression 30x^2 - 24x - 54 is (5x - 9)(6x + 6).