Lesson 16

This warm-up prompts students to evaluate the kinds of numerical expressions they will see in the lesson. The expressions involve rational square roots, fractions, and the \(\pm\) notation.

As students work, notice any common errors or challenges so they can be addressed during the class discussion.

Tell students to evaluate the expressions without using a calculator.

Students may be unfamiliar with evaluating rational expressions in which the numerator contains more than one term. To help students see the structure of the expressions, consider decomposing them into a sum of two fractions. For example, show that \(\frac{8\pm2}{5}\) can be written as \(\frac85\pm\frac25\). This approach can also help to avoid a common error of dividing only the first term by the denominator (\(\frac{4+7}{2}\ne2+7\)). Some students may incorrectly write \(\frac{\sqrt{36}}{2}\) as \(\sqrt{18}\). Point out that the first expression is equal to 3 while the other has to be greater than 3 since \(\sqrt{18}\approx4.243\).

Select students to share their responses and reasoning. Address any common errors. As needed, remind students of the properties and order of operations and the meaning and use of the \(\pm\) symbol.

In this activity, students encounter equations that are challenging to solve using the methods they have learned, motivating students to seek a more efficient method.

Arrange students in groups of 2. Ask partners to choose the same equation. Give students quiet time to solve the equation and then time to discuss their solutions and strategy. If they finish solving their chosen equation, ask them to choose another one to solve. Leave a few minutes for a whole-class discussion.

Provide access to calculators for numerical computations.

Consider arranging students who solved the same equation in groups of 2 to 3 to discuss their strategies and then displaying the correct solutions for all to see.

Invite students to share their reflections on the solving process. Discuss questions such as:

• “What method did you choose and why?”
• “Did you find yourself choosing one method and then switching to another? If so, what prompted you to do that?”
• “Did you run into any challenges when rewriting the equation (or completing the square)? What were some of the challenges?”

Acknowledge that all of these equations are cumbersome to solve by either rewriting in factored form or completing the square. The last equation cannot be written in factored form (with rational coefficients), so completing the square is the only way to go. Tell students they are about to learn a formula that gives the solutions to any quadratic equation.

This activity introduces the quadratic formula. Students begin by applying the formula and using it to solve various equations, ranging from those that can be easily solved using other methods to the kinds that would be quite tedious to solve without the formula. They then verify that the formula gives the same solutions as those calculated by another method. Students notice that, for equations that cannot be easily rewritten using factored form or solved by completing the square, the formula offers quite an efficient way to find the solutions.

Display the equation \(ax^2 + bx + c =0\) for all to see. Tell students that  \(a\), \(b\), and \(c\) are numbers and \(a\) is not 0. Ask students if they could complete the square for this equation without first replacing \(a\), \(b\), and \(c\) with numbers.

Explain that this can indeed be done! The outcome of completing the square is not going to be numerical solutions (because no numbers are used), but rather a general formula for finding the solutions of the quadratic equation. While students won’t have to complete the square for this equation now, they will see the formula and try using it. 

Display the quadratic formula for all to see. Tell students that when an equation is of the form \(ax^2 + bx + c =0\), where \(a\), \(b\), and \(c\) are numbers and \(a\) is not 0, we can find its solutions by using the formula: \(\displaystyle x=\dfrac{\text- b \pm \sqrt{b^2-4ac}}{2a}\)

Guide students through the steps of using the formula to find the solutions to \(x^2 - 8x + 15=0\).

• “First, what do we expect the solutions to be?” (3 and 5, because we can rewrite it as \((x-3)(x-5)=0\).)
• “If \(x^2 - 8x + 15\) is \(ax^2 + bx + c\), what are the values of \(a\), \(b\), and \(c\)?” (\(a=1,b=\text-8,c=15\).)
• Write out the formula as is.

• Replace \(a\), \(b\), and \(c\) with the corresponding numbers from the equation:

  \(\displaystyle x=\dfrac{\text- (\text-8) \pm \sqrt{(\text-8)^2-4(1)(15)}}{2(1)}\)

• Evaluate one part of the expression at a time, ending with \(x = 3\) and \(x=5\).

Provide access to calculators for numerical computations.

If time is limited, ask students to complete at least 2 equations, including an equation in which the leading coefficient is not 1.

Much of the student discussion will have happened in small groups. Focus the whole-class conversation on whether the quadratic formula works for solving all equations and when it might be a preferred method. Ask students,

• “Look at the list of equations and the work you did to solve them. Do certain equations lend themselves to certain methods of solving? Why?” (We can see the factors of some expressions right away, so rewriting it in factored form would be the quickest method to solve the equations. Other expressions cannot be easily rewritten in factored form, but can be easily transformed into perfect squares. For example, when the coefficient of \(x^2\) is 1, the coefficient of the linear term is an even number, and the constant term is also a whole number, completing the square is straightforward. When the squared term has a coefficient other than 1, the other two methods are less practical, so the quadratic formula may be the quickest approach.)
• “Why do you think the quadratic formula is useful only when the \(a\) in \(ax^2+bx+c=0\) is not 0?” (If \(a\) is 0, the expression is no longer quadratic because the squared term disappears, leaving only \(bx+c\), which is linear.)

Select students who used the quadratic formula to solve the last few equations to explain their solutions and display their work for all to see. Discuss any challenges or disagreements in using the formula.

Tell students that they will use the formula to solve other equations and find out more about its merits and how it compares to other methods of solving.