Aim: How do we solve linear-quadratic systems algebraically?

The way we solve linear-quadratic systems algebraically is by making the two quadratic equations equal to each other.
Example:
  y=2x+8
  y=x^2+x-12
You combine both equations.
  2x+8=x^2+x-12
Then you get both equations to one side by either adding or subtracting to make the whole equation equal to zero.
  0=x^2-x-20
You do the diamond problem to find to multiples that add up to -1 and that multiplies to -20
  0=(x-5)(x+4)
  x=5, x=-4
You then plug back in both numbers for what you got for x but separately into the equations to find what y equals. Then your final answers: (5,18), (-4,0)
  

Aim: How do we solve linear-quadratic systems of equation?

The way we solve linear-quadratic systems is by combing the two equations where both Y's are equal to each other.
Example:
  y=6x
  y=4x-8
You make these to equations equal to each other:
  6x=4x-8
Then you want to get the X's by itself to find what x equals. You do this by subtracting 4x on both sides.
  2x=-8
Then you divide 2 on both sides to get what X equals.
  x=-4
You then plug in what you got for X in either equations.
y=6(-4)
y=-24
That's your solution: (-4,-24)

Aim: How do we solve quadratic equations by taking square roots?

If b^2-4ac is a perfect square, then the two roots are rational.
If b^2-4ac isn't a perfect square, then the two roots are irrational.
Examples:

y=x^2+6x+8

a= x^2
b=6
c=8
b^2-4ac= 6^2-4(1)(8)
36-4(8)
36-32=4
Rational b/c 4 is a perfect square

y=x^2=3x-1
a=1
b=3
c=-1
b^2-4ac= 3^2-4(1)(-1)
9-4(-1)
9+4=13
Irrational b/c 13 isn't a perfect square

Aim: How do we use the discriminant to find the number of solutions to a quadratic equation?

The discriminant is: b^2-4ac
If b^2-4ac is greater than 0, then you'll get two solution's.
If b^2-4ac is equal to 0, then you'll get one solution.
If b^2-4ac is less than 0, then you'll get no solutions.

When you have a quadratic equation and your using the discriminant, you first find what a, b, and c equal.
Quadratic equation example: x^2+6x+9=0
A equals: x^2 of a quadratic equation
B equals: 6x
C equals: 9
You then use the discriminant: b^2-4ac, to find how many solutions the quadratic equation has.