Lesson 16

The Quadratic Formula

16.1: Evaluate It (5 minutes)

Warm-up

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.

Launch

Tell students to evaluate the expressions without using a calculator.

Student Facing

Each expression represents two numbers. Evaluate the expressions and find the two numbers.

  1. \(1 \pm \sqrt{49}\)
  2. \(\displaystyle \frac{8 \pm 2}{5}\)
  3. \(\pm \sqrt{(\text-5)^2-4 \boldcdot 4 \boldcdot 1}\)
  4. \(\displaystyle \frac{\text-18 \pm \sqrt{36}}{2 \boldcdot 3}\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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 \(\sqrt{18}\approx4.243\).

Activity Synthesis

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.

16.2: Pesky Equations (10 minutes)

Activity

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.

Launch

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.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support small-group discussion. Provide the class with the following sentence frames to help them discuss their solution strategies with their partner: "First I _____ because. . .”, "I tried _____ and what happened was . . .”, “How did you get . . . ?”, and “Why did you . . . ?” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning when solving quadratic equations.
Design Principle(s): Support sense-making
Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. Activate prior knowledge by reminding students that they have already successfully completed tasks by both factoring and completing the square. Invite them to brainstorm what might be important to consider when selecting a method to use.
Supports accessibility for: Social-emotional skills; Conceptual processing

Student Facing

Choose one equation to solve, either by rewriting it in factored form or by completing the square. Be prepared to explain your choice of method.

  1. \(x^2-2x-1.25=0\)
  2. \(5x^2+9x-44=0\)
  3. \(x^2+1.25x=0.375\)
  4. \(4x^2-28x+29=0\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

16.3: Meet the Quadratic Formula (20 minutes)

Activity

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.

Launch

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.

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 4 questions to complete. Chunking this task into more manageable parts may also support students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

Here is a formula called the quadratic formula.

\(\displaystyle x=\dfrac{\text- b \pm \sqrt{b^2-4ac}}{2a}\)

The formula can be used to find the solutions to any quadratic equation in the form of \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are numbers and \(a\) is not 0.

This example shows how it is used to solve \(x^2 - 8x + 15=0\), in which \(a=1\), \(b=\text-8\), and \(c=15\).

\(\displaystyle \begin {align} x &=\dfrac{\text- b \pm \sqrt{b^2-4ac}}{2a} &\qquad &\text{original equation}\\ x &=\dfrac{\text- (\text-8) \pm \sqrt{(\text-8)^2-4(1)(15)}}{2(1)} &\qquad &\text{substitute the values of }a, b, \text{and }c &\\ x &=\dfrac{8 \pm \sqrt{64-60}}{2} &\qquad &\text {evaluate each part of the expression} \\ x &=\dfrac{8 \pm \sqrt{4}}{2} \\x &=\dfrac{8 \pm 2}{2}\\ x &=\frac{10}{2} \qquad \text {or} \qquad x =\frac{6}{2} \\x &=\text{ }5 \qquad \text{ } \text {or} \qquad x =\text{ }3 \end{align}\)

Here are some quadratic equations and their solutions. Use the quadratic formula to show that the solutions are correct.

  1. \(x^2 + 4x - 5 = 0\). The solutions are \(x=\text-5\) and \(x=1\).
  2. \(x^2 + 7x + 12 = 0\). The solutions are \(x=\text-3\) and \(x=\text-4\).
  3. \(x^2 + 10x + 18 = 0\). The solutions are \(x={\text-5} \pm \frac{\sqrt{28}}{2}\).
  4. \(x^2 - 8x + 11 = 0\). The solutions are \(x=4 \pm\frac{ \sqrt{20}}{2}\).
  5. \(9x^2-6x+1=0\). The solution is \(x=\frac13\).
  6. \(6x^2+9x-15=0\). The solutions are \(x=\text-\frac52\) and \(x=1\).

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. Use the quadratic formula to solve \(ax^2+c=0\). Let’s call the resulting equation P.
  2. Solve the equation \(3x^2-27=0\) in two ways, showing your reasoning for each:

    • Without using any formulas.
    • Using equation P.
  3. Check that you got the same solutions using each method.

  4. Use the quadratic formula to solve \(ax^2+bx=0\). Let’s call the resulting equation Q.
  5. Solve the equation \(2x^2+5x=0\) in two ways, showing your reasoning for each:

    • Without using any formulas.
    • Using equation P.
  6. Check that you got the same solutions using each method.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. At the appropriate time, give students 2–3 minutes to plan what they will say when they share their responses to the question, “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?” in the activity synthesis. Encourage students to consider what details are important to share and to think about how they will explain their reasoning using mathematical language. Invite students to rehearse what they will say to the whole class with their partner. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

Display the equation \(25x^2-50x+16=0\) for all to see. Ask students how they prefer to solve it (by rewriting the expression in factored form, completing the square, or using the quadratic formula) and why. Then, ask them to solve the equation using their preferred method.

Possible explanations for the different methods:

  • Rewriting in factored form: The equation can be rewritten as \((5x-8)(5x-2)=0\) and solved using the zero product property.

  • Completing the square: \(25x^2-50x+25\) is a perfect square, so we can just add 9 to either side of the original equation and find the square root of 9.

  • The quadratic formula, because it always works.

The solutions (by any method) are \(\frac25\) and \(\frac85\).

16.4: Cool-down - Solving and Checking (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

We have learned a couple of methods for solving quadratic equations algebraically:

  • by rewriting the equation as \(\text{factored form}=0\) and using the zero product property
  • by completing the square

Some equations can be solved quickly with one of these methods, but many cannot. Here is an example: \(5x^2-3x-1=0\). The expression on the left cannot be rewritten in factored form with rational coefficients. Because the coefficient of the squared term is not a perfect square, and the coefficient of the linear term is an odd number, completing the square would be inconvenient and would result in a perfect square with fractions.

The quadratic formula can be used to find the solutions to any quadratic equation, including those that are tricky to solve with other methods.

For an equation of the form \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are numbers and \(a \neq 0\), the solutions are given by:

\(\displaystyle x=\dfrac{\text- b \pm \sqrt{b^2-4ac}}{2a}\)

For the equation \(5x^2-3x-1=0\), we see that \(a=5\), \(b=\text-3\), and \(c=\text-1\). Let’s solve it!

\(\displaystyle \begin {align} x &=\dfrac{\text- b \pm \sqrt{b^2-4ac}}{2a} &\qquad &\text{the quadratic formula}\\ x &=\dfrac{\text-(\text-3) \pm \sqrt{(\text-3)^2-4(5)(\text-1)}}{2(5)} &\qquad &\text{substitute the values of }a, b, \text{and }c &\\ x &=\dfrac{3 \pm \sqrt{9+20}}{10} &\qquad &\text {evaluate each part of the expression} \\ x &=\dfrac{3 \pm \sqrt{29}}{10}\end{align}\)

A calculator gives approximate solutions of 0.84 and -0.24 for \(\frac{3 + \sqrt{29}}{10}\) and \(\frac{3 - \sqrt{29}}{10}\).

We can also use the formula for simpler equations like \(x^2-9x+8=0\), but it may not be the most efficient way. If the quadratic expression can be easily rewritten in factored form or made into a perfect square, those methods may be preferable. For example, rewriting \(x^2-9x+8=0\) as \((x-1)(x-8)=0\) immediately tells us that the solutions are 1 and 8.