1. Factoring

Zero Product Theorem  If the product of two factors is zero then at least one of the two factors must be zero.

Example

x(x - 1) = 0

implies that either

x = 0     or     x = 1

A Review of Factoring

Done by working several examples such as

x2 - 3x - 10 = (x - 5)(x + 2)

For a complete review of factoring go to Factoring and for an interactive tutorial on the AC method for factoring go to http://mathcsjava.emporia.edu/~greenlar/AC/AC.html

The Square Root Method

If the middle term of a quadratic is zero then the best way to solve is to isolate the x2 term and then take square roots of both sides.

Example

4x2 - 9 = 0         Subtract 9 from both sides to produce

4x2 = 9               Divide by 4

x2  = 9/4             take the square root

x = +- 3/2

Caution:  Don't forget that you get two solutions- the "plus" and the "minus" solution.

2. Completing the Square

For a full review of completing the square go to Completing the Square for an interactive tutorial on completing the square go to http://mathcsjava.emporia.edu/~greenlar/CompleteTheSquare/CompleteTheSquare.html

Here is a brief explanation by example:

2x2 - 8x + 2

1. Factor the leading coefficient:  2(x2 - 4x + 1)

2. Calculate -b/2:  -(-4)/2 = 2

3. Square the solution above:  22  = 4

2(x2 - 4x + 4 - 4 + 1)

5. Regroup:  2[(x2 - 4x + 4 ) - 3]

6. Factor the inner parentheses using part two as a hint:  2[(x - 2)2 - 3]

7. Multiply out the outer constant:    2(x - 2)2 - 6

8. Breath a sigh of relief.

Recall (otherwise memorize) the quadratic formula: The solutions to

ax2  + bx + c = 0

are and We will use this often, so memorize it.  Recall that the discriminant

D = b2  - 4ac

is a convenient measure of determining how many roots (solutions) there are.  We have:

 D Number of Roots Positive 2 Negative 0 zero 1

Example