In this lesson, students begin to solve quadratic equations by reasoning about what values would make the equations true and by using structure in the equations. The idea that some quadratic equations have two solutions is also made explicit. Students may begin to record their reasoning process as steps for solving, but this is not critical at this point, as it will be emphasized in a later lesson.

As students reason about an equation, they may intuitively perform the same operation on each side of the equal sign to get closer to the solution(s). When they reach equations of the form , it is important to refrain from telling students to “take the square root of each side.” Instead, focus on reasoning about values that would make the equation true (MP2).

Reasoning this way helps to curb two common misconceptions:

• The only solution to an equation such as is . Each positive number has two square roots, one positive and the other negative.
• Squaring is invertible.

When solving an equation such as , these notations are commonly used to express the solutions:

•  or  
•  and  

The use of “or” is really a shorthand for "If is a number such that , then or .” The use of “and” is a shorthand for “Both and are values that make the equation true." Either notation can be appropriate, depending on how the question is stated.

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. Consider making technology available.

Solving the problems in the lesson gives students many opportunities to engage in sense making and perseverance (MP1). They also make use of the structure of quadratic equations to reason about solutions (MP7).