Completing the Square - Maple Programming Help

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Completing the Square

Main Concept

Completing the square is the name of a process used to convert quadratic polynomials in the general form to the vertex form:

ax2 +bx+c axh2 +k 

where

h= b2 a,    k = cb24 a

Steps:

1. Factor the leading coefficient out of the first two terms

a x2+b x+c  =

ax2 +bax +c

2. Complete the square by adding and subtracting the "magic number"

 

(the square of half the coefficient of x)

=

ax2+ bax  + b2 a2  b2 a2 +c

3. Move the constant b2 a2 outside the parentheses. Remember to multiply it by a.

=

ax2+ bax  + b2 a2  +c  b24 a

4. Factor the perfect square and add the remaining terms.

=

ax+ b2 a 2+c  b24 a

Numerical Example:

1. Factor the leading coefficient out of the first two terms

3 x2+6 x+4  =

3x2 +2 x +4

2. Complete the square by adding and subtracting the "magic number"

 

(the square of half the coefficient of x)

=

3x2+ 2 x  + 1  1 +4

3. Move the constant 1 outside the parentheses. Remember to multiply it by 3.

=

3x2+ 2 x +1+4 3

4. Factor the perfect square and add the remaining terms.

=

3x+ 12+1

 

Click "Next Step" to follow the steps of completing the square. Click "New Quadratic" to start from a different polynomial. Observe that the magic number is the square of half the coefficient of x.

 

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