Lesson 3.4 Completing the Square

Transcription

1 Lesson 3. Completing the Square Activity 1 Squares of Binomials 1. a. Write a formula for the square of a binomial: ÐB :Ñ œ Notice that the constant term of the trinomial is coefficient of the linear term (or B -term) is., and the b. Now we'll reverse the process. Add a constant to the expression that will turn it into a perfect square: B 'B œ ÐB Ñ (Do you see how to find the constant? Here's what you do:) First find : À : œ ', so : œ Ð 'Ñ œ Then find : À : œ We can turn any quadratic expression into a perfect square by adding the correct constant. This process is called completing the square.. Complete the square and write the result as the square of a binomial. a. B )B œ b. B %B œ For parts ( c) and ( d), do not use a calculator! Work with fractions. c. B $B œ & d. B B œ 55

2 Activity We can visualize completing the square using rectangles. 1. Study the diagram that illustrates completing the square for B!B : Step 1 Represent B!B as a rectangle Step Take of the linear term, and rearrange the pieces Step 3 What is the missing piece? Step B!B & œ ÐB &Ñ. In the space below, draw a similar set of diagrams to illustrate completing the square for B 'BÞ Activity 3 Solving Quadratic Equations by Completing the Square We use the technique of completing the square to solve quadratic equations. There are two parts to this method: 1.. Create a perfect square Use extraction of roots 1. a. Follow the steps to solve B 'B ' œ 0 Move the constant term to the other side of the equation: B 'B œ ' 5

3 Complete the square on the left side: : œ ' œ : œ $ œ ( ) and ( ) Add * to both sides of the equation: B 'B * œ ' * Write the left side as the square of a binomial: Ð Ñ œ & Use extraction of roots to find the solutions; take the square root of both sides: B $ œ The solutions are and. b. Graph the parabola C œ B 'B ' on your calculator, and copy the graph onto the grid. What are the B-intercepts of the graph? or or B $ œ Solve B )B % œ! by completing the square. (This equation cannot be solved by factoring!) Move the constant term to the right side: Complete the square: : œ Ð )Ñ œ : œ Add : to both sides: Write the left side as the square of a binomial, and simplify the right side: Extract roots to obtain two solutions: Use a calculator to find decimal approximations for each solution. Round to thousandths: 57

4 3. Our method for completing the square only works if the coefficient of B is. If the lead coefficient is not 1, we must divide each term of the equation by the lead coefficient. a. Solve %B B œ 0 by completing the square. Divide each term by %: Move the constant to the right: Complete the square on the leftà : œ Ð $Ñ œ, : œ Add : to both sides: Write the left side as a perfect square; simplify the right side. Extract roots to obtain two solutions: Use a calculator to find decimal approximations for each solution. Round to thousandths: b. Graph C œ %B B in the standard window. Sketch the graph on the grid. What are the B-intercepts of the graph?

5 Wrap-Up In this Lesson, we worked on the following skills and goals related to quadratic models: Add a constant to B,B to create a perfect square Solve quadratic equations by completing the square Check Your Understanding 1. Name three algebraic methods for solving a quadratic equation.. Give an example of a quadratic trinomial that is the square of a binomial. 3. In Activity 3, what were the two parts named for solving an equation by completing the square?. What is wrong with the following solution? B )B $ œ! B )B ' œ $ ' ÐB %Ñ œ $ % È$ 59