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Algebra Practice Exam

In groups of about 4, complete each of the following 5 parts. You need not do them in order, but you should
do part 5 last. (If you are less familiar with foiling and factoring, I suggest starting with step 1 and going
through in order . You should have one piece of paper per group with solutions to the questions for each

Part I
Algebra Blocks

Send up one representative to get the algebra blocks and get a quick demo of how they are used.
Multiply the following binomials together using the algebra blocks. (Use the L-shaped piece. Choose one of
the unknow lengths for x and line up the parts of each binomial, one per side. Now ll in the center with
pieces that match each side.)

Now, use the blocks the other way, factor the following quadratics. Grab enough pieces to have the equations
and arrange them in a rectangle. Choose two adjacent sides and see what the factors are.

Part II
Factoring by grouping

One method of factoring quadratics (without guess and check) relies on factoring by grouping. Here is an
example of factoring by grouping.

The method goes like this . Make sure your quadratic is written in the form ax2 + bx + c. Note what a, b,
and c are equal to. Now find the product ac. Find two factors of the number ac that add up to b. Rewrite
your b term as the sum of those factors. Finally factor by grouping.

Here is an example.

5x2 + 11x + 2 So a = 5, b = 11, and c = 2
ac = 5· 2= 10 So my product of ac is 10
10 ·1 = 10 and 10 + 1 = 11 So I choose 10 and 1 to be my factors of ac that add up to b
5x2 + 11x + 2 = 5x2 + 10x + 1x + 2
  Now that I have rewritten 11x as 10x + 1x I factor by grouping
5x(x + 2) + 1(x + 2) = (5x + 1)(x + 2)

Thus I can factor 5x2 + 11x + 2 as (5x + 1)(x + 2). Now you try some.

1. 4x2 + 7x - 15 (Hint: Since c is negative, one of the factors of ac will be positive, the other will be
negative )
2. x2 + 5x + 4
3. x2 + 8x + 15
4. x2 - 7x + 12 (Hint: Since b is negative and c is positive, both factors of ac will be negative)

Part III
The Zero-Product Property

Let A and B be some unknow expressions. If A ·B = 0 what do I know about A and B? Well, if the product
is 0, then either A had to be zero or B had to be 0. This is known as the Zero-Product Property. If we
have to solve a quadratic equation (that can be factored), we will use this property to find our solutions.
For example:

Do you see how I used the Zero-Product Property on the 3rd line? Now you try solve some equations using
the Zero-Product Property.

Part IV
The Quadratic Formula

If I have a quadratic written in the form ax2 +bx+c = 0, then I can use the quadratic formula to find what
x is. The quadratic formula isAn example: Solve 2x2 + 3x - 1 = 0. So here I have

a = 2, b = 3, c = -1. Now I substitute these into my formula as follows:

I cannot simplify my expression any more, so my solutions are and
There is a song that can be used to memorize that can be used to memorize the quadratic formula. It is to
the tune of "Pop goes the weasel" and can be found if you rummage around my website. Find it and come
show me where it can be found to get credit for doing so. Now you try some:

1. 2x2 + 9x - 5 = 0
2. 2x2 = 4x + 1 (Hint: Be sure to write this in the form ax2 + bx + c = 0 before applying the formula.)
3. 9x2 - 12x - 5 = 0
4. x2- x = 14 (Hint: See hint on 1 in this section.)

Part V
Wrapping it all up

It's go time! Solve the following equations using the method of your choice.

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