What do we already know about solving quadratic equations?
• Understand the Zero Product Property and use it to justify steps to solve a factorable quadratic equation (A-REI.A.1).
• Explore forms of quadratic equations that can be solved by seeing structure (A-SSE.A.2, A-REI.B.4b).
• Connect solving quadratic equations to finding zeros of quadratic functions (A-SSE.B.3A$^\star$, F-IF.C.8a$^\star$).
In the Quadratic Functions unit students saw that the factored form of a quadratic function makes it easy to see the zeros of the function. Here they see that fact in a different light, learning that finding the zeros of a quadratic function is the same as solving the quadratic equation obtained by setting the function equal to zero. Students use factoring as an opportunistic method to solve quadratic equations, taking advantage of situations where a factorization is readily available from seeing the structure of a quadratic expression (e.g. a difference of squares) (MP7). Then they see how the vertex form of a quadratic function can also be useful in solving quadratic equations by taking square roots, as preparation for the general method of completing the square in the next section.
Note: The fact that a number has at most two square roots is connected to the factored form. For example, the fact that $x^2=4$ has only the two solutions $2$ and $–2$ follows from the fact that the equation $x^2-4=0$ can be converted to factored form $(x+2)(x-2)=0.$
Tasks
1
Zero Product Property 1
WHAT: The series of tasks
Zero Product Property 1, Illustrative Mathematics (https://www.illustrativemathematics.org/content-standards/tasks/2141)
Zero Product Property 2, Illustrative Mathematics (https://www.illustrativemathematics.org/content-standards/tasks/2142)
Zero Product Property 3, Illustrative Mathematics (https://www.illustrativemathematics.org/content-standards/tasks/2143)
Zero Product Property 4, Illustrative Mathematics (https://www.illustrativemathematics.org/content-standards/tasks/2144)
leads students to understand the Zero Product Property and apply it to solving quadratic equations. The emphasis is on seeing the structure of a factored or factorable expression and justifying the steps in a solution (MP7), rather than memorizing steps without understanding. First, students state and prove the Zero Product Property. Then they use it to find solutions of equations in factored form. Finally, they encounter equations that they must rewrite in factored form before they can use Zero Product Property to solve them.
WHY: In this unit we treat factoring as an opportunistic method for solving quadratic equations, to be used when a factorization is readily available. Reasoning with this technique requires understanding the Zero Product Property. Understanding the connection between factors and values of the variable that make an expression equal zero helps students draw connections between factors and zeroes in quadratic and polynomial functions. Polynomials with degree higher than two may be encountered incidentally in this unit, but will be the focus of study in a later unit.
2
Solving a Simple Cubic Equation
WHAT: Students solve the equation $9x = x^3$ by factoring, graph $f(x) = x^3 – 9x$, explain the connection between the equation and the graph, and then investigate the consequences of the move “dividing both sides by x”.
WHY: In the previous unit, students learned that the factored form of a quadratic function makes it easy to see the zeros of the function. This task shows that fact in a different light; finding the zeros of a quadratic function is the same as solving the quadratic equation obtained by setting the function equal to zero. Additionally, students learn of a common pitfall in solving more complicated equations – that “doing the same thing to both sides” is not okay if the move is not invertible. In this case, it is not okay to divide both sides of the original equation by x, because it disregards the possibility that x is zero.
3
Quadratic Sequence 1
WHAT: In the previous unit, students created equivalent expressions for the number of squares in visual patterns in a sequence. These equivalent expressions define a quadratic function, giving the number of squares as a function of the step in the sequence. In this activity they find which pattern in a sequence yields a certain number of squares. So they must find the input value that makes the quadratic function have a given output value, that is, they must solve a quadratic equation. They solve these equations by algebraic reasoning.
WHY: These activities connect solving quadratic equations with previous work on sequences of visual patterns. Students see that sometimes it is possible to get a certain number of squares, because the corresponding quadratic equation has a positive integer solution, and that at other times it is not.
4
Quadratic Sequence 2
WHAT: In the previous unit, students created equivalent expressions for the number of squares in visual patterns in a sequence. These equivalent expressions define a quadratic function, giving the number of squares as a function of the step in the sequence. In this activity they find which pattern in a sequence yields a certain number of squares. So they must find the input value that makes the quadratic function have a given output value, that is, they must solve a quadratic equation. They solve these equations by algebraic reasoning.
WHY: These activities connect solving quadratic equations with previous work on sequences of visual patterns. Students see that sometimes it is possible to get a certain number of squares, because the corresponding quadratic equation has a positive integer solution, and that at other times it is not.