to change a quadratic equation from vertex form to general form to learn to square a binomial and...

• To change a quadratic equation from vertex form to general form

• To learn to square a binomial and factor perfect-square expressions using rectangle diagrams

• To solve problems using a quadratic equation that models projectile motion

•You have learned two forms of a quadratic equation. ▫The vertex form: y=a(x-h)2+k, gives

you information about transformations of the parent function, y=x2.

▫The general form: y=ax2+bx+c, is used to model many projectile motion situations.

MONOMIAL BINOMIAL

TRINOMIAL POLYNOMIAL

Here are some examples of polynomials.

•Is each algebraic expression a polynomial? If so, combine like terms and statehow many terms it has. If not, give a reason why it is not a polynomial.

NO NO

NO YES

YESYES

Sneaky SquaresThere are many different, yet equivalent, expressions for a number. For example, 7 is the same as 3+4 and as 10-3. In this investigation you will use these equivalent expressions to model squaring binomials with rectangle diagrams.•This diagram shows how to express 72 as (3+4)2.•Find the area of each of the inner rectangles. What is the sum of the rectangular areas? What is the area of the overall square? What conclusions can you make?

•For each expression below, draw a diagram on your Rectangle Diagram Template like the one on the previous slide. Label the area of each rectangle and find the total area of the overall square.

• Even though lengths and areas are not negative, you can use the same kind of rectangle diagram to square an expression involving subtraction. You can use different colors, such as red and blue, to distinguish between the negative and the positive numbers. For example, this diagram shows 72 as (10-3)2.

•Draw a rectangle diagram representing each expression. Label each inner rectangle and find the sum.

•You can make the same type of rectangle diagram to square an expression involving variables.

•Draw a rectangle diagram for each expression. Label each inner rectangle and find the total sum. Combine any like terms you see and express your answer as a trinomial.

•Now use what you have learned to create a rectangle diagram for a trinomial.

•Make a rectangle diagram for each trinomial. In so doing, what must you do with the middle term? Label each side of the overall square in your diagram, and write the equivalent expression in the form (x-h)2.

•Use your results from the previous step to solve each new equation symbolically.

•Remember, quadratic equations can have two solutions.

x2 3x

3x 9

49

2

2 2

6 9 49

3 7

3 7

3 7

4 10

x x

x

x

x

X or

=

•Numbers like 49 are called perfect squares because they are the squares of integers, in this case 7 or -7. The trinomial x2+6x+9 is (x+3)2. So it is also called a perfect square.

•Which of these trinomials are perfect squares?

•Explain how you can recognize a perfect-square trinomial when the coefficient of x2 is 1. What is the connection between the middle term and the last term?

•Square the expression (x-h)2 by making a rectangle diagram. Then describe a shortcut for this process that makes sense to you.

Example B•Rewrite y=2(x+3)2-5 in the general form,

y = ax2 + bx + c.