So far, the quadratic expressions that students have transformed from standard form to factored form have at least a squared term and a linear term. In this lesson, students encounter quadratic expressions without a linear term and consider how to write them in factored form.
Students begin by studying numerical examples and noticing that expressions such as \((20+1)(20-1)\) and \(20^2-1^2\) (which is a difference of two squares) are equivalent. Through repeated reasoning, students are able to generalize the equivalence of these two forms as \((x+m)(x-m) =x^2-m^2\) (MP8). Then, they make use of the structure relating the two expressions to rewrite expressions (MP7) from one form to the other.
Along the way, they encounter a variety of quadratic expressions that can be seen as differences of two squares, including those in which the squared term has a coefficient other than 1, or expressions that involve fractions.
Students also consider why a difference of two squares (such as \(x^2 - 25\)), can be written in factored form, but a sum of two squares (such as \(x^2+25\)) cannot be, even though both are quadratic expressions with no linear term.
After this lesson, students will have the tools they need to solve factorable quadratic equations given in standard form by first rewriting them in factored form. That work begins in the next lesson.
- Understand that multiplying a sum and a difference, $(x+m)(x-m)$, results in a quadratic with no linear term and explain (orally) why this is the case.
- When given quadratic expressions with no linear term, write equivalent expressions in factored form.
- Let’s look closely at some special kinds of factors.
- I can explain why multiplying a sum and a difference, $(x+m)(x-m)$, results in a quadratic expression with no linear term.
- When given quadratic expressions in the form of $x^2+bx+c$, I can rewrite them in factored form.
In an algebraic expression, the coefficient of a variable is the constant the variable is multiplied by. If the variable appears by itself then it is regarded as being multiplied by 1 and the coefficient is 1.
The coefficient of \(x\) in the expression \(3x + 2\) is \(3\). The coefficient of \(p\) in the expression \(5 + p\) is 1.
In an expression like \(5x + 2\) the number 2 is called the constant term because it doesn't change when \(x\) changes.
In the expression \(5x-8\) the constant term is -8, because we think of the expression as \(5x + (\text-8)\). In the expression \(12x-4\) the constant term is -4.
The linear term in a quadratic expression (In standard form) \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, is the term \(bx\). (If the expression is not in standard form, it may need to be rewritten in standard form first.)
zero product property
The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0.