Quadratic Equations

Things one must know:

• Quadratic equations are equations with one unknown and two defining components:
  1) a variable term raised to the second power
  2) a variable term raised to the first power
• Factoring quadratic equations:
  1) Move all the terms to the left side of the equation, combine them, and put them in the form ax² + bx + c (where a, b, and c are integers). The right side should equal 0.
  2) Factor the equation: find two integers whose sum equals b and whose product equals c.
  3) Rewrite the equation in the form (x + ?)(x + ?), where the question marks represent the two integers you solved for in the previous step.
  4) Since this product equals 0, one or both of the factors must be 0. Set each factor independently to 0 and solve for x.  The two solutions for x have the opposite signs of the integers we found in step three.
• If you have a quadratic expression equal to 0, and you can factor an x out of the expression, then x = 0 is a solution of the equation. Be careful not to just divide both sides by x, this eliminates the solution x = 0. You are only allowed to divide by a variable (or any expression) if you are absolutely sure that the variable or expression does not equal zero.
• FOIL - First, Outer, Inner, Last
• Some quadratics have only one solution, these are called perfect square quadratics.
• When zero appears in the denominator of an expression, then that expression is undefined. Making the the denominator of the fraction equal 0 would NOT make the entire expression equal zero.
• Three common quadratic expressions:
  #1 x² - y² = (x + y)(x - y)
  #2 x² + 2xy + y² = (x + y)(x + y) = (x + y)²
  #3 x² - 2xy + y² = (x - y)(x - y) = (x - y)²
• If the other side of the equation is a perfect square quadratic, the problem can be quickly solved by taking the square root of both sides of the equation. Be sure to consider both the positive and the negative square root. 
• You can substitute a variable to create a quadratic.
• Determining the number of solutions a quadratic has: b² - 4ac
  If the result is greater than zero, there will be two solutions. If it is equal to zero, there will be one solution. If it is less than zero, there will be no solutions.