In this activity students predict which equations have real solutions and which ones do not.

Press students to give reasons for their predictions rather than guessing. There are many valid methods of making a prediction about whether there are real solutions. Monitor for students who use the following strategies, listed in order of efficiency for a general quadratic equation, with real coefficients.

• Try to factor the expression to find any real solutions
• Sketch a graph of the function to see if it has -intercepts
• Use the quadratic formula or completing the square to find the actual solutions
• Strategically use the quadratic formula to determine whether the discriminant,  , is negative and conclude that the solutions will be real only if it's not negative

In the following activity, it will be helpful for students to have a reliable and efficient method for generating quadratic functions that have real or non-real zeros. Factoring is inefficient and it could be difficult for students to use factoring to generate functions that have non-real zeros. Looking at the function's graph is a more reliable way to see whether it has real zeros, but graphing technology is not always readily available and some functions may appear to touch the -axis when they do not. Checking the sign of the discriminant is a comparatively quick and reliable test for whether a function has real zeros, and this strategy can also be used to create a function that will or will not have real zeros.

The purpose of this discussion is to articulate that the discriminant can reveal the number and type of solutions for a quadratic equation.

Invite previously selected students to share their predictions and reasoning. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “Which strategies are most efficient for finding the number and type of solution in general?” (Using the discriminant is fairly quick to do and applies to any quadratic equation in the form .)
• “Are other strategies more efficient when the quadratic equations are in particular forms?” (Yes, sometimes it is easy to find the actual solutions. For example, when the equation is in forms like , , or .)
• “How might you use these strategies to write a quadratic equation with 2 non-real solutions?” (I could invent values for , , and so that is negative.)

The goal of this activity is for students to use the strategies they have learned to build their own quadratic equations with either real or non-real solutions. While guess-and-check is possible, the emphasis here should be on strategic use of the quadratic formula or completing the square. For example, in an earlier lesson students saw that a quadratic such as has non-real solutions when  .

Monitor for students who use the strategy of choosing , , and so that will be either negative or not and for students using ideas from completing the square.

If students are not yet comfortable using the discriminant or other algebraic techniques to create their equations, consider saying:

• “Can you explain how you created your equation.”

• “Can you write an equation in the form that has the number and type of solutions you want? How can you rewrite that equation to be in standard form?”

Invite 2–3 previously identified pairs to share one of their equations, how they knew what type of solutions the equation has, and how they found the solutions.

Then invite students to share observations about how the equations with real solutions are different from the ones with non-real solutions. Depending on the specific equations students wrote, patterns may or may not be apparent. This is why it's helpful to have a quick test like checking the sign of the discriminant to figure out whether a quadratic equation has real solutions.