Algebra: Chapter 6-3, p 270
Trinomial Squares
We are going backward again, using the properties:
`(a + b) * (a + b) = a^2 + 2ab + b^2 = (a + b)^2`
`(a – b) * (a – b) = a^2 – 2ab + b^2 = (a – b)^2`
For a trinomial square to factor, we must make sure that:
- 2 of the terms must be squares, `a^2` and `b^2`
- There must be NO MINUS sign before the `a^2` and `b^2`
- If we multiply `a` and `b` and double the result, we get the 3rd term, `2ab` or its additive inverse `−2ab`.
Sometimes, we can also factor out a coefficient in front of the `a^2`, like `2a`. We MIGHT be able to factor out the 2 before we start the trinomial determination.
Here is a purplemath link that describes trinomial squares, scroll down about 1/2 way to get to the information.
Math-8, Chapter 8-8, p 412
Systems of Equations
Solving Systems of Equations by Graphing
A set of equations for which a common solution is called a “system of equations”. A “solution” of a system of 2 equations in 2 variables is an ordered pair that makes both equations true. When we find all the solutions of a system, we say that we have solved the system.
To determine if a given point (x, y) is a solution to the system of equations, we use the (x, y) points in both equations to see if they evaluate to true or false. If BOTH equations are true, then the point (x, y) is a solution.
To determine a given point [that is to solve the (x, y) points], of 2 equations, we can do this graphically by graphing both linear equations on ONE graph. Where the 2 lines cross or intersect becomes the solution to the system of equations.
REMEMBER, we can plot the equations with a T-charts using MrE’s favorite points for x, (0, 1, and 2) or we can use the equation in the form: `y=mx+b`, where m is the slope and b is the y-intercept (0, y). HINT, the slope and y-intercept approach is easier!
By the way, when doing this, we can come up with 3 things that may happen:
- The lines have 1 point of intersection. The point is the ONLY solution of the system
- The lines are parallel. If this is true, the system has NO solutions because NO point (x,y) is common to both linear equations.
- The lines coincide or lie on top of each other. Since they are the same line, there are INFINITE numbers of solutions.
Here is a link from purplemath.com for both types of problems.