Today we mainly focused on quadratic expression factorization (因式分解). This is a skill that is very useful in quickly solving quadratic equations. While it is not hard, it does take practice to get good at. So we spent most of the time letting the students going exercises.

Basically, for the simple form of x^2 + bx + c (that is, the coefficient in front of x^2 is 1) to be able to be factored as (x + p) (x + q), then p + q must be equal to b and pq (p times q) must be equal to c. So the basic steps are, first look at the factors for c and find two whose product is c, then check if their sum equals to b, if so, you have it. If not, try different pairs.

Then for the general form of ax^2 + bx + c (a <> 0), it is the same idea but a little more complicated. For it to be factored into (px + m)(qx + n), then pq (p times q) must be equal to a, mn( m times n) must be equal to c and pn + qm must be equal to b. The basic steps are, first look at the factors for a and pick two so that its product is a, then pick two factors of c so that its product is c, then check if the cross multiplication of those two pairs is b.

A cross multiplication chart will help

px         +m        | +pnx

X             |

qx          +n        | + qmx

—————————————-

pqx^2 + mn   | (pn + qm)x

With this skill, we looked at solving quadratic equations where we factor first.

Note, we have not got into the general solution of a quadratic equation yet.

Homework:

Page 19 18 a c e g i k m  o, 19 b d f h j l n, 20 a ii, f ii, g ii, i ii, j ii, k ii

We talked fractional equations and word problems with fractional equations.

Any equation involving any fraction, either x (unknown)  in enumerator or denominator, is called a fractional equation, or also called rational equation.

We studied fractional equations with x in enumerators, and we studied equations with x in denominators. When x is in denominators, one important step is to check the answer to make sure that it does not make any denominator zero and throw away any answer that does.

For word problems, it is important to come up with the right equation with the given information.

Homework is as follows:

Page 9: 49 a c d e g i k m n o, 52 a, 53, 54, 55, 56

We covered the following material today:

• Scientific Notation or Standard Form
• Special Algebraic Rules for Calculations
• Expanding Products

To represent a number, scientists use the form of A * 10^n where 1 <= A < 10, n is an integer. This is called scientific notation or standard form. The key point is, A must be smalled than 10 and not smaller than 1.

We talked about how to convert from a decimal form to the standard form, and vice versa. We talked about how to perform addition, subtraction, multiplication, and division with two numbers in standard form.

For special algebraic rules, we established the following three rules:

• (a + b)^2 = a^2 + 2ab + b^2
• (a – b)^2 = a^2 -2ab + b^2
• (a^2 – b^2) = (a + b)*(a – b)

For expanding products, the key point we went over is the distributive law, using which one should be able to expanding products in any form.

For homework, it is on the Workbook:

• Page 4: 11b, 11d, 11f, 12d, 12f, 12h, 15a, 15b, 15c, 15d
• Page 5: 18a, 18b
• Page 6: 1b, 1d, 2c, 2d, 3d, 3f
• Page 7: 4a, 4c, 4e, 5c, 5e, 5f

We covered the classroom first. With 25 students in the class, we need everyone’s cooperation to have a good learning environment for everyone.

We covered the material about Exponents. The following are the important rules and definitions to memorize:

• a^m * a^n = a^(m+n)
• a^m / a^n  = a^(m-n)
• (a^n)^m = a^(nm)
• a^m * b^m = (ab)^m
• a^m / b^m = (a/b)^m
• a^0 = 1, a<> 0 (a to the power of zero is 1, as long as a is not zero)
• a^(-n) = 1 / a^n, a <> 0

With the above rules and defintions, we can simply expressions. We did several in-class exercises and the homework will be mostly about simplying expressions.

We also talked about exponents with variables, whose operation is no different, except that we use variables, like x, y, a, b, c, etc., to represent the things in the operation. Same rules and definitions apply.

For homework, it is on the workbook:

• Page 1, 1a, 1c, 1e, 1g, 2l, 2n, 2p, 2r
• Page 2, 4a, 4b, 4c, 6a, 6b, 7a, 7b, 7c
• Page 3, 8g, 8i, 9a, 9c