Lesson 7

In an earlier lesson, students transformed quadratic expressions from standard form into factored form. There, the factored expressions are products of two sums, \((x+m)(x+n)\), or two differences, \((x-m)(x-n)\). Students continue that work in this lesson, extending it to include expressions that can be rewritten as products of a sum and a difference, \((x+m)(x-n)\).

Through repeated reasoning, students notice that when we apply the distributive property to multiply out a sum and a difference, the product has a negative constant term, but the linear term can be negative or positive (MP8). Students make use of structure as they take this insight to transform quadratic expressions into factored form (MP7). They see that if a quadratic expression in standard form (with coefficient 1 for \(x^2\)) has a negative constant term, one of its factors must have a negative constant term and the other must have a positive constant term.