Linear and Quadratic Equations

The 3 special equations

(x-y)(x+y)=x^2-y^2 (x+y) (x+y)=x^2+2xy+y^2 (x-y) (x-y)= x^2-2xy+y^2

N variables N equations Myth

A problem may involve two variables but only one equation and you would still be able to solve it. If x>y, x and y are positive integers, and x+y=3, then x=? X would clearly be 2 and Y would be 1.

N variables N equations Myth

Don't assume that are sufficient to determine the values of two variable ESPECIALLY when one equation is just a multiple of the other equation. What is the value of xy? 1. x+y=2 2. 10x+10y=20 Both of these statements are same. Once you reduce statement 2.

factoring by grouping

Example: x^3-x^2-16x+16=0

Be cautious when an expression is set equal to zero

INCORRECT: x (x+100)=0 x+100=0 x=-100 CORRECT: x(x+100)=0 x=0 x=-100

An expression or equation with common factor, the common factor can be factored out

If 20rxy + 20sxy = 100 and r+s=5, what is the value of xy? 1. Factor out the 20xy. 20xy (r+s) =100 2. That's 20xy (5) =100. 3. xy has to equal 1.

When one equation can solve for.two variable example

If x and y are positive integers, and 5x + 8y=55, then x=? 8y=55-5x We can factor out the 5s. 8y=5(11-x) Next, we can isolate y by dividing both side of the equation by 8: 5(11-x)/8 We know that (11-x) has to be divisible by 8 so x could equal to 3. 40/8=5

What's a linear equation? What a quadratic equation?

Linear: 10x+20=40 Quadratic: x^2 +10x +5

Strategy: Spotting the difference of squares quickly

Look for the square of value minus the square of another value. x^2-9= (x-3) (x+3) because x^2 is the square of x and 9 is the square of 3.

If x and y are positive integers, what is the sum of x and y? 1. 5x+9y=63 2.xy=18

Statement 1 is sufficient. 5x=63-9y 5x=9(7-y) 5x/5=9(7-y)/5 y must equal to 2 because 7-2=5. If we have y then we have x.

Another way to express -1

x-y/y-x=-1