5 - Solving Quadratic Equations

In the last unit, students used multiple representations to make sense of quadratic functions and solve problems with tables and graphs. In this unit, students solve quadratic functions using what they learned from the last unit and begin to use algebraic strategies to solve quadratic equations. Students should identify which of these strategies is most efficient for any given situation. Additionally, when using algebraic strategies, students will produce real and imaginary/complex solutions. However, a more formal study of operations with imaginary/complex numbers will be done in Algebra 2.

As students use algebraic strategies to find real or imaginary/complex solutions to quadratic equations, they can use what they know about transforming quadratic functions to see if their solutions makes sense. Similarly, students can use what they know about key features of quadratics (e.g., symmetry) to analyze or justify their answers.

The use of algebra tiles or area models provide opportunities for students to generate and make sense of algebraic strategies like completing the square and factoring. For example, as students repeatedly convert quadratics in standard form (represented with tiles) to vertex form, they see a pattern from the concrete model that leads to the procedure for completing the square. Once students understand the concept of completing the square, they can use it to make sense of the quadratic formula. (See an example of this process by using the following link.

https://www.youtube.com/watch?v=tHhO1_Snpsw)

In unit 2, students solved systems of linear equations both approximately, with tables and graphs, and exactly using algebraic strategies. In unit 3, students found approximate solutions for systems of equations consisting of both linear and exponential equations. In this unit, students extend those experiences to again find both approximate and exact solutions for systems that include linear, exponential and quadratic equations.

There are standards listed in this section for two reasons.

1. The standards have been modified to be appropriate for this unit. Text in gray font is part of the Michigan K-12 standard but does not apply to this unit. Text in brackets denotes a modification that has been made to the standard.
2. The standards contain content that is developed and/or utilized across multiple units.

Modified For this Unit

The Complex Number System

HSN-CN.A. Perform arithmetic operations with complex numbers.

HSN-CN.A.1. Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.

Creating Equations

HSA-CED.A. Create equations that describe numbers or relationships.

HSA-CED.A.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Reasoning with Equations and Inequalities

HSA-REI.D. Represent and solve equations and inequalities graphically.

HSA-REI.D.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational,absolute value, exponential, and [quadratic] logarithmic functions.

Developed and/or Utilized Across Multiple Units

Seeing Structure in Expressions

HSA-SSE.A. Interpret the structure of expressions.

HSA-SSE.A.2. Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ – y⁴ as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²).

Reasoning with Equations & Inequalities

HSA-REI.D. Represent and solve equations and inequalities graphically.

HSA-REI.D.10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Building Functions

HSF-BF.A. Build a function that models a relationship between two quantities.

HSF-BF.A.1. Write a function that describes a relationship between two quantities.

HSF-BF.A.1a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

1. What is the relationship between the number of real roots and the graph of a quadratic equation? Why does this relationship exist?
2. How does paying attention to the parameters of a quadratic function and its graph assist in finding solutions?
3. What are strategies for solving quadratic equations? When might one strategy be more efficient than another?
4. How is solving a system consisting of a linear or a non linear equation and a quadratic equation similar/different to solving linear systems and/or quadratic equations?