Up to this point, most quadratic expressions that students have transformed from standard form to factored form had a leading coefficient of 1, that is, they were in the form of \(x^2 + bx +c\) because the squared term had a coefficient of 1. There were a few instances in which students rewrote expressions in standard form with a leading coefficient other than 1. Those expressions were differences of two squares, where there were no linear terms (for instance, \(9x^2-64\) or \(25x^2-9\)). Students learned to rewrite these as \((3x+8)(3x-8)\) or \((5x+3)(5x-3)\), respectively.
In this lesson, students consider how to rewrite expressions in standard form where the leading coefficient is not 1 and the expression is not a difference of two squares. They notice that the same structure used to rewrite \(x^2 + 5x+4\) as \((x+4)(x+1)\) can be used to rewrite expressions such as \(3x^2+8x+4\), but the process is now a little more involved because the coefficient of \(x^2\) has to be taken into account when finding the right pair of factors. The work here gives students many opportunities to look for and make use of structure (MP7).
This lesson aims to give students a flavor of rewriting more complicated expressions in factored form, and to suggest that it is not always practical or possible. This experience motivates the need for other strategies for solving equations and prepares students to complete the square in a series of upcoming lessons.
- Given a quadratic expression of the form $ax^2+bx+c$, where $a$ is not 1, write an equivalent expression in factored form.
- Write a quadratic equation that represents a context, consider different methods for solving it, and describe (orally) the limitations of each method.
- Let’s transform more-complicated quadratic expressions into the factored form.
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)
- I can use the factored form of a quadratic expression or a graph of a quadratic function to answer questions about a situation.
- When given quadratic expressions of the form $ax^2+bx+c$ and $a$ is not 1, I can write equivalent expressions 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.