Algebra: Chapter 6-4, p 273
Factoring `x^2 + bx + c`
In the polynomial `x^2 + bx + c`, remember that `c` is called the constant term or just the constant. If the constant of a poynomial `x^2 + bx + c` is NOT a perfect square, the trinomial CANNOT be factored into a square of a binomial. It is possible to factor it as the product of 2 different binomials.
Using the X (the cross) METHOD, we can find the factors if we first understand that these are quadratics of the form `ax^2 + bx + c`. We need to find the coefficients a, b and c. At the top of the X, we put the product `ac`, at the bottom of the X, we put the `b`.
For these problems in lesson 6-4, most of the time, `a = 1`.
Being a detective, we need to find 2 numbers that have a product of `ac` and a sum of `b`. Once you have those, the answers become the 2 answers on the left and right of the X.
The CROSS METHOD comes in quite handy and I would ENCOURAGE all students to try it out. IT WILL MAKE LIFE MUCH EASIER FOR YOU! Here is a link from Purplemath, it too is pretty good at having examples!
Math-8, Chapter 8-9, p 418
Graphing Inequalities
Inequalities in 2 variables (e.g., `2x + y > 5`) can be solved just like the equation `2x + y = 5`. The only difference is that the solutions lie in a 1/2 plane instead of on the line. The trick to to find what 1/2 plane they satisfy. To solve these equations, we can use a T-chart or solve for the x-intercept (set `y = 0`) and the y-intercept (set `x = 0`) ordered pairs. Plot them on the graph and connect the dots!
Definitions:
- Half-planes: Regions above the line and below the line (or to the left or right), there are 2 half-planes.
- Boundary line: The line in an equation is renamed to this in an inequality.
If the inequality possesses a < or >, then the line is dashed (or dotted) to show that values ON the line DO NOT satisfy the inequality.
If the inequality possesses a ≤ or ≥, then the line is SOLID to show that values on the line SATISFY the inequality.
To determine what 1/2 plane to shade, test the ordered pair (0,0) [or (1,1) if that is on the line]. If (0,0) is true for the inequality, then we shade that portion’s [where the (0,0) lies] 1/2 plane as the solution space. If it is NO true, then we shade the opposite 1/2 plane.