Algebra

f(x) = 6x^2 + 19x + 8 Star method

Step-by-step solution with explanation

Final Answer

Factored form: ; zeros are and

Step-by-step solution

1

Identify a, b, and c values

In a quadratic , we label each coefficient. We need these for the Star (AC) method.
2

Multiply a and c together

The Star method starts by multiplying the first and last coefficients. We need two numbers that multiply to 48 AND add to 19 (our b value).
3

Find the Star pair of numbers

We hunt for a factor pair of 48 that adds up to 19. The pair 3 and 16 works perfectly.
4

Rewrite the middle term using the pair

We split the middle term into using our Star pair. The expression is still equal — we just rewrote it.
5

Group terms and factor each group

Group the first two and last two terms. Factor out the GCF from each group. Both groups share the factor , which confirms we chose the right pair.
6

Factor out the common binomial

Since both groups contain , we pull it out front. What remains is . This is the fully factored form.
7

Set each factor equal to zero

To find the zeros of , set each factor to zero and solve. A quadratic always has up to two solutions, and we found both.

Understanding this problem

Learning Insight

The Star (AC) method works because of the distributive property in reverse. When you split the middle term into two parts that multiply to , you create groups that always share a common binomial factor. It's basically organized reverse-FOIL.

Quick Tip

After finding your Star pair, always check: do they multiply to AND add to ? Both must be true before you move on. Takes 5 seconds and saves a lot of erasing.

Common Mistake

Students often forget to check the sign when finding the Star pair. For example, they find numbers that multiply to 48 but don't notice the signs don't add up to exactly +19. Always verify the sum, not just the product.