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

Verified step by step guidance

1. Identify the trinomial in the form \( ax^2 + bx + c \), where \( a = 9 \), \( b = 4 \), and \( c = -2 \).
2. Multiply \( a \) and \( c \) to get \( 9 \times (-2) = -18 \).
3. Find two numbers that multiply to \(-18\) and add to \(4\). These numbers will be used to split the middle term.
4. Rewrite the trinomial by splitting the middle term \(4x\) using the two numbers found in the previous step.
5. Factor by grouping: Group the terms into two pairs and factor out the greatest common factor from each pair, then factor out the common binomial factor.

Key Concepts

Factoring Trinomials

Factoring trinomials involves rewriting a quadratic expression in the form ax^2 + bx + c as a product of two binomials. This process requires identifying two numbers that multiply to ac (the product of a and c) and add to b. Understanding this concept is essential for simplifying expressions and solving quadratic equations.

Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of a quadratic equation. It is particularly useful when factoring is difficult or impossible. Knowing how to apply this formula can help verify the factors of a trinomial and find its solutions.

Discriminant

The discriminant, given by the expression b² - 4ac, determines the nature of the roots of a quadratic equation. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root; and if negative, there are no real roots. Understanding the discriminant helps in predicting whether a trinomial can be factored over the real numbers.