Free Algebra Tutorials!
Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

## Complete Squares

Look at the following quadratic expression:

x - 4x + 4 = (x - 2)(x - 2) = (x - 2) .

For the obvious reason, this expression is called a complete square. Quadratic expressions which can be factored into a complete square are useful in many situations. They have a particularly simple structure and it is important to be able to recognise such factorizations.

Example 3

Show that the following quadratic expressions are complete squares.

(a) x + 6 x + 9 , (b) x + 4 x + 4 , (c) x - 2 x + 1 , (d) x - 2 ax + a 2 .

Solution

In each of these cases it is easy to check the following.

(a) x + 6 x + 9 = ( x + 3) ,

(b) x + 4 x + 4 = ( x + 2) ,

(c) x - 2 x + 1 = ( x - 1) ,

(d) x - 2 ax + a = ( x - a) .

The last example, x - 2ax + a = (x - a), is a general case and it may be used to find perfect squares for any given example. It may be usefully employed in finding solutions to the following exercises.