In this video lesson, students return to some quadratic functions they have seen. They write quadratic equations to represent relationships and use the quadratic formula to solve problems that they did not previously have the tools to solve (other than by graphing). In some cases, the quadratic formula is the only practical way to find the solutions. In others, students can decide to use other methods that might be more straightforward (MP5).
The work in this lesson—writing equations, solving them, and interpreting the solutions in context—encourages students to reason quantitatively and abstractly (MP2).
In this video lesson, students apply what they learned about transforming expressions into factored form to make sense of quadratic equations and persevere in solving them (MP1). They see that rearranging equations so that one side of the equal sign is 0, rewriting the expression in factored form, and then using the zero product property make it possible to solve equations that they previously could only solve by graphing. These steps also allow them to easily see—without graphing and without necessarily completing the solving process—the number of solutions that the equations have.
Previously in this video series, students used area diagrams to expand expressions of the form (x + p)(x + q) and generalized that the expanded expressions take the form of x2 + (p + q)x + pq. In this video lesson, they see that the same generalization can be applied when the factored expression contains a sum and a difference (when p or q is negative) or two differences (when both p and q are negative).
Students transition from thinking about rectangular diagrams concretely, in terms of area, to thinking about them more abstractly, as a way to organize the terms in each factor. They also learn to use the terms standard form and factored form. When classifying quadratic expressions by their form, students refine their language and think about quadratic expressions (MP6).
In this video lesson, students contrast visual patterns that show quadratic relationships with those that show linear and exponential relationships. To analyze the patterns, students generate tables of values, write expressions, and create graphs. They also encounter the term quadratic expression and learn that a quadratic relationship can be written using an expression with a squared term.
This video lesson builds on the idea that both graphing and rewriting quadratic equations in the form of expression = 0 are useful strategies for solving equations. It also reinforces the ties between the zeros of a function and the horizontal intercepts of its graph, which students began exploring in an earlier unit.
Here, students learn that they can solve equations by rearranging them into the form expression = 0, graphing the equation y = expression, and finding the horizontal intercepts. They also notice that dividing each side of a quadratic equation by a variable is not reliable because it eliminates one of the solutions. As students explain why certain maneuvers for solving quadratic equations are acceptable and others are not, students practice constructing logical arguments (MP3).
Previously in this video series, students saw that a squared expression of the form (x + n)2 is equivalent to x2 + 2nx + n2. This means that, when written in standard form ax2 + bx + c (where a is 1), b is equal to 2n and c is equal to n2. Here, students begin to reason the other way around. They recognize that if ax2 + bx + c is a perfect square, then the value being squared to get c is half of b, or (b/2)2. Students use this insight to build perfect squares, which they then use to solve quadratic equations.
Students learn that if we rearrange and rewrite the expression on one side of a quadratic equation to be a perfect square, that is if we complete the square, we can find the solutions to the equation.
Rearranging parts of an equation strategically so that it can be solved requires students to make use of structure (MP7). Maintaining the equality of an equation while transforming it prompts students to attend to precision (MP6).
Earlier in this video series, students transformed quadratic expressions from standard form into factored form. There, the factored expressions are products of two sums, (x + m)(x + n), or two differences, (x – m)(x – n). Students continue that work in this video lesson, extending it to include expressions that can be rewritten as products of a sum and a difference, (x + m)(x – n).
Through repeated reasoning, students notice that when we apply the distributive property to multiply out a sum and a difference, the product has a negative constant term, but the linear term can be negative or positive (MP8). Students make use of the structure as they take this insight to transform quadratic expressions into factored form (MP7).
In this video lesson, students encounter quadratic expressions without a linear term and consider how to write them in factored form. Through repeated reasoning, students are able to generalize the equivalence of these two forms: (x + m)(x – m) and x2 – m2 (MP8). Then, they make use of the structure relating the two expressions to rewrite expressions (MP7) from one form to the other. Students also consider why a difference of two squares (such as x2 – 25) can be written in factored form, but a sum of two squares (such as x2 + 25) cannot be, even though both are quadratic expressions with no linear term.
This video lesson has two key aims. The first aim is to familiarize students with the structure of perfect-square expressions. Students analyze various examples of perfect squares. They apply the distributive property repeatedly to expand perfect-square expressions given in factored form (MP8). The repeated reasoning allows them to generalize expressions of the form (x + n)2 as equivalent to x2 + 2nx + n2.
The second aim is to help students see that perfect squares can be handy for solving equations because we can find their square roots. Recognizing the structure of a perfect square equips students to look for features that are necessary to complete a square (MP7), which they will do in a future video lesson.
In this video lesson, students encounter the quadratic formula and learn that it can be used to solve any quadratic equation. They use the formula and verify that it produces the same solutions as those found using other methods, but can be much more practical for certain equations.
Using the quadratic formula to solve equations requires students to attend carefully to the parameters in the given equations (MP6) and to apply different properties of operations flexibly as they reason symbolically (MP2).
This video lesson serves two main purposes: to reiterate that some solutions to quadratic equations are irrational, and to give students the tools to express those solutions exactly and succinctly. Students recall that the radical symbol (√) can be used to denote the positive square root of a number. Many quadratic equations have a positive and a negative solution, and up until this point, students have been writing them separately. Here, students are introduced to the plus-minus symbol (±) as a way to express both solutions. Students also briefly recall the meanings of rational and irrational numbers. They see that sometimes the solutions are expressions that involve a rational number and an irrational number—for example, x = ±√8 + 3. Students make sense of these solutions by finding their decimal approximations and by solving the equations by graphing. The work here gives students opportunities to reason quantitatively and abstractly (MP2).
In this video lesson, students learn that completing the square can be used to solve any quadratic equation, including equations that involve rational numbers that are not integers. Students notice that the process of completing the square is the same when the equations involve messier numbers as when they have simple integers, but the calculations may be more time consuming and prone to error. An error-analysis activity highlights some common errors related to completing the square.
Completing the square for quadratic expressions that are more elaborate encourages students to look for and make use of the same structure that helped them when they were working with less complicated expressions (MP7).
