Some students may wonder why they need to both add and subtract a number from the standard form expression in order to complete the square. In previous lessons, students completed the square while solving an equation, often adding the same number to each side of the equal sign to maintain equality. Here, there is no equal sign. Emphasize that each move must generate an equivalent expression. Show a few expressions such as 5, , , , , and . Ask them which ones are equal to 5. Next, ask them to write two more expressions that include the number 5 and also equal 5. Encourage students to notice that the sum of the numbers excluding 5 must be 0.

Invite students to share their graph and response to the last question. Emphasize that while graphing is a quick way to check whether two expressions define the same function, it is not always reliable. If we make an algebraic error, the graph would almost certainly show it. But having two graphs that appear to be identical does not prove that two expressions are indeed equivalent. We can only be sure that two expressions define the same function by showing equivalence algebraically. For example, when students convert the expression in vertex form back to standard form, does it produce the original expression?

Make sure students understand that whatever operation is performed on an expression to complete the square, it should not change the value of the expression. Adding opposites (for example, 9 and -9 or -25 and 25), or adding and subtracting the same number, has the effect of adding 0, which keeps the original and the transformed expressions equivalent.

Alternatively, the constant term can be rewritten as a sum or difference of two numbers that have the same value as the constant term and include the constant needed to complete the square. For example, in the equation from the Launch.

Select students to display their responses for all to see. Discuss questions such as:

• “How did you know what factor to use to rewrite the original expression?” (Find a factor that all terms in an expression have in common. Use the coefficient of the squared term so that, after the expression is rewritten, the squared term has a coefficient of 1.)
• “In , both 2 and 4 are common factors. Is one number better than the other for our purposes here?” (4 is a better factor to use because it allows the squared term to have a coefficient of 1, which makes it easier to complete the square. If we use 2, the squared term still has an inconvenient coefficient that is neither 1 nor a perfect square.)
• “How can we check if the expression in vertex form is equivalent to the original expression?” (One way is to convert it back into standard form and see if it is the same expression as the original.)

After students have completed their work, discuss the correct answers to the questions and any difficulties that came up.

Highlight for students that different forms of quadratic expressions are useful in different ways, so it helps to be able to move flexibly between forms. For example, a quadratic expression in factored form makes it straightforward to determine the zeros of the function that the expression defines and the -intercepts of its graph. The vertex form makes it easy to identify the coordinates of the vertex of the graph of the function.

To know whether two expressions define the same function, we can rewrite the expression in an equivalent form. There are many tools at our disposal. For instance, we can rewrite an expression in factored form, apply the distributive property to expand a factored expression, rearrange parts of an expression, combine like terms, or complete the square.