Students may incorrectly think that can represent a different value in each factor in an equation. For example, upon finding -11 and 3 as solutions to , they think that one solution is for the in and the other for the in .

Remind students that solving the equation is like finding the zeros of the function defined by . Although there may be two values of that lead to 0 for the value of , only one input can be entered into the function at a time. Ask students to substitute the solutions into the equations and check if the expression is equal to 0 each time.

• When , the value of the expression is , or , which is 0.
• When , the value of the expression is , or , which is 0.

Invite students to share their strategies for solving the nonlinear equations. As they explain, record and organize each step of their reasoning process, and display for all to see.

For example, the equation tells us that if the product of and is 0, then either is equal to 0 or is equal to 0. We can then organize the rest of the solving process as:

The equation is true when or when .

Emphasize that because at least one of the factors must be 0 for the product to be 0, we can set each expression that is a factor equal to 0 and solve each of these equations separately.

Remind students that we can check our solutions by substituting each one back into the equation and seeing if the equation remains true. Although the two factors, and , won’t be 0 simultaneously when 3 or is substituted for , the expression on the left side of the equation will have a value of 0 because one of the factors is 0.

• When is 3, the expression is , or , which is 0.
• When is , the expression is , or , which is 0.

This activity enables students to apply the zero product property to solve a contextual problem and reinforces the idea of solving quadratic equations as a way to reason about quadratic functions.

Previously, students have encountered two equivalent quadratic expressions that define the same quadratic function. Here, they work to show that two quadratic expressions—one in standard form and the other in factored form—really do define the same function.

Monitor for these likely strategies, and select students who use various strategies to share in the discussion. Students may:

• Graph both equations on the same coordinate plane and show that they coincide.
• Inspect a table of values of both equations and show that the same output results for any input.
• Use the distributive property to multiply the expression in factored form to show that .

Next, they consider whether the standard form or factored form better helps them find the zeros of the function. They then use that form to find the zeros without graphing. The work here reiterates the connections between finding the zeros of a quadratic function and solving a quadratic equation where a quadratic expression that defines a function has a value of zero.

Ask students to share their responses and reasoning. Discuss questions such as:

• “Why is the factored form more helpful for finding the time when the object has a height of 0 meters?” (To find the input values when the output has a value of 0 is to solve the equation . When the expression is in factored form, we can use the zero product property to find the unknown inputs.)
• “What if we tried to solve the equation in standard form by performing the same operation on each side?” (We would get stuck. For instance, we could add or subtract terms from each side, but then there are no like terms to combine on either side, so we are no closer to isolating the variable.)

If no students related solving equations in factored form to using the factored form to find the horizontal intercepts of a graph of a quadratic function, discuss this connection.

• “In an earlier unit, we saw that the factored form of a quadratic expression, such as , allows us to see the -intercepts of its graph, but we didn’t look into why the graph crosses the -axis at those points. Can you explain why it does now?” (The -intercepts have a -value of 0, which means the quadratic function is 0 at those -values: . If multiplying two numbers gives 0, one of them must be 0. So either or . If , then is 5. If , then is -9.)