Unit 6 Solving Quadratic Equations

Lesson 1

Learning Focus

Explore the meaning of a fraction as an exponent.

Lesson Summary

In this lesson, we developed meaning for using a fraction as an exponent. We began with a geometric sequence and found a way to fit data at the points halfway between whole number inputs in a way that would maintain the multiplicative behavior of the sequence. We continued to work with different fractional inputs and gave meaning to using a variety of fractions as exponents.

Lesson 2

Learning Focus

Examine how the properties of exponents work with rational exponents.

Write equivalent exponential functions using different growth factors.

Lesson Summary

In this lesson, we continued to explore the meaning of rational exponents, including negative integer exponents and fractional exponents. We learned that the properties of exponents can be applied to all rational exponents, not just integer exponents.

Lesson 3

Learning Focus

Change the form of radical expressions using properties of exponents.

Lesson Summary

In this lesson, we learned how to change the form of complicated radical and exponential expressions using the properties of radicals and exponents. Strategies for changing the form of radical expressions can be explained by converting the radical expressions to exponential form.

Lesson 4

Learning Focus

Solve quadratic equations graphically and algebraically.

Make connections between solving quadratic equations and graphing quadratic functions.

Lesson Summary

In this lesson, we learned methods for solving quadratic equations. Some quadratic equations can be solved using inverse operations and taking the square root of both sides of the equations. Some quadratic equations can be solved by factoring and using the zero product property. Some quadratic equations can be solved by completing the square and then using inverse operations. Quadratic equations that have real solutions can also be solved by graphing, and each of these algebraic methods has connections to graphing.

Lesson 5

Learning Focus

Understand and use a formula for solving quadratic equations.

Lesson Summary

In this lesson, we found a formula by completing the square for solving quadratic equations. We used the quadratic formula to find exact and approximate solutions to quadratic equations and connected those solutions to the graphs of quadratic functions. We learned to rewrite exact solutions so that fractions do not contain common factors in the numerator and denominator and the square roots do not contain factors that are perfect squares.

Lesson 6

Learning Focus

Solve quadratic equations efficiently and accurately.

Solve systems of quadratic and linear equations.

Lesson Summary

In this lesson, we compared methods for solving quadratic equations. We found that some equations lend themselves to one method, and other equations are more efficiently solved with other methods. Using technology to graph is always a useful way to check solutions.

Lesson 7

Learning Focus

Write quadratic functions in vertex, factored, and standard form.

Find roots of a quadratic function.

Use the roots of a quadratic function to write the function in factored form.

Lesson Summary

In this lesson, we examined solutions to quadratic equations and connected them with the graph of the function. Solutions for a quadratic equation can be used to write the function in factored form in a process that is the reverse of solving an equation by factoring. We found that when the graph of the quadratic function did not cross the $x$ -axis, the quadratic formula gave solutions that included the square root of a negative number. Although there is not a real number that can be squared to get a negative number, these expressions seemed to work like square roots. Since the Fundamental Theorem of Algebra predicts that all quadratic functions will have two roots, the nature of solutions that involve the square root of a negative number still needs to be resolved.

Lesson 8

Learning Focus

Relate irrational numbers to physical quantities such as the hypotenuse of a right triangle.

Understand expressions that contain negative numbers inside a square root, like $\sqrt{- 1}$ .

Add, subtract, and multiply complex numbers.

Lesson Summary

In this lesson, we connected irrational numbers to the measure of geometric figures and showed where a given irrational number is located on the number line. We found irrational solutions of quadratic equations and used the solutions to write the equation in factored form. We also learned of a new set of numbers defined in terms of $\sqrt{- 1} = i$ and $i^{2} = - 1$ . We examined how these numbers fit into the number system and performed arithmetic operations on them.

Lesson 9

Learning Focus

Support or challenge claims about different types of numbers and the result of adding, subtracting, multiplying, and dividing.

Lesson Summary

In this lesson we examined claims about sets of numbers and the operations of addition, subtraction, multiplication, and division. An example of such a claim is: The quotient of two whole numbers is always a whole number. A counterexample that shows this claim to be false is: $2 \div 7 = \frac{2}{7}$ . Since $\frac{2}{7}$ is a rational number, this example shows that dividing two whole numbers does not always result in a whole number.

Lesson 10

Learning Focus

Solve quadratic inequalities both graphically and algebraically.

Interpret solutions to quadratic inequalities that arise from context.

Lesson Summary

In this lesson, we developed a strategy for solving quadratic inequalities. The procedure involves solving the related quadratic equation and then using the graph or testing values to find the intervals that are solutions to the inequality. If the inequality represents a real context, the solutions must be interpreted so that they fit the situation.