Factoring Quadratic Expressions

Objective: Find two integers with a specified sum and product.

Prerequisite skills: integer products and polynomial expansion. To factor x² + bx + c into (x + r)(x + s), require r + s = b and rs = c. A positive product requires same-sign factors, while the sum determines which sign. Common error: matching the product but not the sum. Restrictions and interpretation: A factored identity must match both the linear coefficient and constant term.

Objective: Use products of binomials when the leading coefficient is not one.

In (ax + r)(bx + s), the leading coefficient is ab, the constant is rs, and the middle coefficient is as + br. All three must match. Splitting the middle term can reveal grouping. Common error: using r + s as the middle coefficient when a and b are not both 1. Restrictions and interpretation: A factored identity must match both the linear coefficient and constant term.

Objective: Remove a common factor before interpreting a quadratic factor.

First extract a common factor, then factor the reduced quadratic. A numerical multiplier does not change the roots, but it must remain in an equivalent expression. This lesson emphasizes expression structure; solving equations follows in chapter 6. Common error: discarding the leading multiplier when rewriting an expression. Restrictions and interpretation: A factored identity must match both the linear coefficient and constant term. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.