Lesson 3

Explore

 

 

Landscape engineering, or just landscaping, is the discipline where science and mathematics are used to shape landscapes. A landscape engineer plans and designs the landscape to meet certain criteria, such as minimizing costs or maximizing green space. Those who hire landscape engineers may also have a specific look in mind. Therefore, the aesthetics of the project are also important. Like other engineers, landscape engineers apply mathematical principles to build landscapes and to overcome anticipated design problems.

 

In this lesson you will extend the concept of completing the square to solving quadratic equations. You will build the foundational skills and understanding that will be important in the next lesson when you develop a formula for solving any quadratic equation.

 

You have learned several ways of solving quadratic equations. You can select an appropriate strategy for solving a quadratic equation based on how the equation is presented. Try the next activity to see if you can find an efficient way of determining the roots of some quadratic equations.

 

Try This 1

1. Consider the quadratic equation (x − 1)2 = 9. There are different ways of solving this equation. Show how you would solve this equation algebraically. How many roots are there?
   
   
2. Consider the equation x2 = 9. How is this equation similar to the equation (x − 1)2 = 9?
   
   
3. Solve the equation x2 = 9 using the same strategy you used to solve (x − 1)2 = 9. How many roots are there?
   
   
4. Describe other ways to solve these equations.

Save your work in your course folder.

 

Share 1

 

With a classmate, compare your solving strategies and solutions for Try This 1. The solutions to the equations should be the same, regardless of the method used.

• If your solutions differ, work together to determine where the algebraic error was made.
  
  
• If your solutions are the same but you used different strategies, try solving the equations using your classmate’s method.
  
  
• If you used the same method and arrived at the same solution, try solving the equations using a completely different method. See if you can reach the same answer you were able to obtain with your original solving strategy.

Did either of you use the following strategies?

• Expand and then factor a trinomial.
  
  
• Rearrange the terms and then factor a difference of squares.
  
  
• Take the square root of each side.

If not, give one or two of these strategies a try.

 

Save any changes you made to the Try This 1 questions in your course folder.

 

Keep your file open for the next part of the lesson.