Lesson 19

Quadratic Steps

These materials, when encountered before Algebra 1, Unit 7, Lesson 19 support success in that lesson.

19.1: Quadratic Error (5 minutes)

Warm-up

In this warm-up, students are asked to spot a common error that often takes the form \(\sqrt{a^2 - b^2} = a - b\). The format of the question here closely resembles the values that students will see when working with the quadratic formula in the associated Algebra 1 lesson.

Student Facing

Here is Han’s work to solve an equation. Determine the error he made and be prepared to explain the correct way to solve it.

\(x= \text{-}3 + \sqrt{3^2-4\boldcdot1\boldcdot2}\)

\(x= \text{-}3+ 3-2\boldcdot1\boldcdot2\)

\(x=\text{-}4\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is to remind students to be careful about common errors when working with square roots. Ask students,

  • “How can you remember where square roots fit into the order of operations?” (I like to think of the expression under the square root as a separate question to evaluate before doing anything else.)
  • “Two common errors occur when using squares and square roots. First, \((a+b)^2 = a^2 + b^2\) is not correct in most cases. Next, \(\sqrt{a^2 + b^2} = a + b\) is also not true in most cases. Choose values for \(a\) and \(b\) to assure yourself that these are not true.” (For the first one, \((2 + 3)^2 = 25\) but \(2^2 + 3^2 = 13.\) For the second one, \(\sqrt{2^2 + 3^2} = \sqrt{13}\) which is about 3.6 but \(2+3=5\).)

19.2: Multiplying to Make Perfect Squares (15 minutes)

Activity

In this activity, students examine values that can be multiplied by a number to create a perfect square. In the associated Algebra 1 lesson, students prove the quadratic formula. As part of the process of proving the quadratic formula, students multiply the equation by \(4a\) so that the coefficient of the quadratic term is a perfect square. This work of this activity supports students in understanding that step.

Student Facing

The class is asked to multiply 5 by a number to make it a perfect square.

  • Jada multiplies the number by 5.
  • Han multiplies the number by 15.
  • Elena multiplies the number by 9.
  • Kiran multiplies the number by 20.
  • Mai multiplies the number by 45.
  1. Do you agree with any of the students that their multiplication will make a perfect square?
  2. Find the pairs of positive integer factors of each of the numbers the students want to use.
  3. What do you notice about the factors of the values that do create a perfect square? What do you notice about the factors of the values that do not create a perfect square?
  4. What are some values you could multiply the number 7 by to make it a perfect square?
  5. Which of these values could be multiplied by a number \(a\) so that the product is a perfect square?
    1. \(a\)
    2. \(3a\)
    3. \(4a\)
    4. \(6a\)
    5. \(9a\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is to explore what kinds of values can be multiplied by a number to get a perfect square. Select students to share their solutions and things they noticed. Ask students,

  • “To complete the square, the coefficient of \(x^2\) must be a perfect square. To make the coefficient of \(x^2\) a perfect square for the equation \(3x^2 - 4x = 5\), what could you multiply the equation by? Explain your reasoning.” (I could multiply by 3, 12, 27, or other values of the form \(3n^2\). That would make the coefficient of \(x^2\) of the form \(9n^2\) which is a perfect square.)
  • “A quadratic equation in standard form looks like \(ax^2 + bx + c = 0\). What does that equation look like when you multiply the equation by \(4a\)?” (\(4a^2x^2 + 4ab + 4ac = 0\))

19.3: Stepping Through Completing the Square (20 minutes)

Activity

In this activity, students go through each step of completing the square. In the associated Algebra 1 lesson, students prove the quadratic formula by completing the square with the general form of a quadratic equation in standard form. By closely examining the steps for an actual example, students should be supported to follow the steps for a more abstract example in the associated Algebra 1 lesson.

Student Facing

For each step of the solution, explain what happened in each step and why that step might be taken.

Solve \(x^2 + 8x - 3 = 6\).

  1. \(x^2 + 8x = 6 + 3\)
  2. \(x^2 + 8x + 16 = 9 + 16\)
  3. \((x+4)^2 = 25\)
  4. \(x + 4 = \pm 5\)
  5. \(x = \text{-}4 \pm 5\)
  6. \(x = 1, \text{-}9\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is to understand every step of solving an equation by completing the square. Select students to share their solutions. Ask students,

  • “How could the first 2 steps be combined?” (Knowing that the left side should have a constant term of 16, you can add 19 to both sides of the equation right at the beginning.)
  • “The way this work is written, the \(\pm\) is kept until the end of the work when the 2 solutions are found. Another option is to write two equations as soon as line 4 of this work. What are the benefits of each method?” (It is less to write if I leave the \(\pm\) until the end. When writing 2 equations, it is easier to remember that there are 2 solutions and the numbers can be combined earlier.)