Factor each trinomial, if possible. See Examples 3 and 4. $9x^2+4x-2$

Verified step by step guidance

1. Identify the trinomial to factor: \$9x^2 + 4x - 2$.
2. Multiply the coefficient of \(x^2\) (which is 9) by the constant term (which is -2), giving \(9 \times (-2) = -18\).
3. Find two numbers that multiply to \(-18\) and add up to the middle coefficient, 4. These numbers will help split the middle term.
4. Rewrite the middle term \$4x$ as the sum of two terms using the numbers found, then group the four terms into two pairs.
5. Factor out the greatest common factor (GCF) from each pair and then factor out the common binomial factor to complete the factoring.

Key Concepts

Factoring Trinomials

Factoring trinomials involves expressing a quadratic expression of the form ax^2 + bx + c as a product of two binomials. This process simplifies the expression and helps solve equations or analyze functions. Recognizing patterns and using methods like trial and error or the AC method are common approaches.

The AC Method

The AC method is a systematic way to factor trinomials when the leading coefficient a is not 1. It involves multiplying a and c, finding two numbers that multiply to ac and add to b, then splitting the middle term accordingly to factor by grouping. This method helps factor complex quadratics efficiently.

Factoring by Grouping

Factoring by grouping is a technique used after splitting the middle term in a trinomial. It involves grouping terms in pairs and factoring out the greatest common factor from each group. If done correctly, the expression can be factored into the product of two binomials, completing the factorization process.