Lesson 6 Comparison Shopping Practice Understanding

Learning Focus

Solve quadratic equations efficiently and accurately.

Solve systems of quadratic and linear equations.

How can you determine the most efficient strategy for solving any particular quadratic equation?

Technology guidance for today’s lesson:

Solve a Quadratic Equation by Graphing: Casio ClassPad Casio fx-9750GIII
Solve Quadratic Systems of Equations by Graphing: Casio ClassPad Casio fx-9750GIII

Open Up the Math: Launch, Explore, Discuss

In this lesson, you will be comparing methods for solving quadratic equations, looking for which method works best for a given equation. The methods you will compare along the way are:

Solving by factoring
Solving by completing the square and using inverse operations
Solving by using the quadratic formula
Solving by graphing

You should also be thinking about polishing your skills in using each of these methods by checking your work and watching for repeated errors, like mixing up the signs or making mistakes in dividing common factors out of fractions. In other words, don’t just work the problems to get them done; work the problems so you’re getting better!

1.

Given: $x^{2} - 2 x - 15 = 0$

a.

Solve by completing the square:

b.

Solve by factoring:

2.

Given: $2 x^{2} + 3 x + 1 = 0$

a.

Solve by factoring:

b.

Solve using the quadratic formula:

3.

Given: $x^{2} + 4 x - 8 = 0$

a.

Solve by completing the square:

b.

Solve using the quadratic formula:

c.

Can this equation be solved by factoring? How can you tell?

4.

Given the system of equations:

$\begin{aligned} y_{1} & = x^{2} - 4 x + 1 \\ y_{2} & = x - 3 \end{aligned}$

a.

Solve by graphing:

b.

Solve algebraically:

Choose any algebraic method, and solve each equation or system of equations. Check your solutions by graphing. Be prepared to share the reason you selected the method you used to solve the equation(s).

5.

$4 x^{2} - 81 = 0$

6.

$2 x^{2} + 4 x = 7$

7.

$(x + 8) (x + 6) = 288$

8.

$\begin{aligned} y_{1} & = x^{2} + x - 2 \\ y_{2} & = - x + 1 \end{aligned}$

Ready for More?

You’ve solved some quadratic equations, and now you have a chance to write your own. Try these challenges.

a.

Write a quadratic equation that has only one solution.

b.

Write a quadratic equation that has two rational solutions that are not integers.

c.

Write a quadratic equation that has no solutions.

Takeaways

Efficient methods for solving quadratic equations in the form $a x^{2} + b x + c = 0$ .

Factoring is an efficient strategy when:

Completing the square is an efficient strategy when:

Quadratic formula is an efficient strategy when:

Graphing is an efficient strategy when:

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.

Retrieval

Find the absolute value of each.

1.

$| 17 |$

2.

$| - 32 |$

3.

$| - 5 - 24 |$

What values would be solutions for the following equations?

4.

$| x | = 34$

5.

$| x + 2 | = 10$

6.

Solve the system of equations.

${\begin{cases} y = 2 x + 16 \\ y = - 7 x + 25 \end{cases}$