Algebra/Quadratic Equation

Algebra
 ← Completing the Square Quadratic Equation Binomial Theorem → 

Derivation edit

The solutions to the general-form quadratic function   can be given by a simple equation called the quadratic equation. To solve this equation, recall the completed square form of the quadratic equation derived in the previous section:

 

In this case,   since we're looking for the root of this function. To solve, first subtract c and divide by a:

 

Take the (plus and minus) square root of both sides to obtain:

 

Subtracting   from both sides:

 

This is the solution but it's in an inconvenient form. Let's rationalize the denominator of the square root:

 

Now, adding the fractions, the final version of the quadratic formula is:

 

This formula is very useful, and it is suggested that the students memorize it as soon as they can.

Discriminant edit

The part under the radical sign,   , is called the discriminant,   . The value of the discriminant tells us some useful information about the roots.

  • If   , there are two unique real solutions.
  • If   , there is one unique real solution.
  • If   , there are two unique, conjugate imaginary solutions.
  • If   is a perfect square then the two solutions are rational, otherwise they are irrational conjugates.

Word Problems edit

Need to pull word problems from http://teachers.yale.edu/curriculum/search/viewer.php?id=initiative_07.06.12_u&skin=h