Cake on Fridays: Completing the Square

Monday, December 20, 2010

Completing the Square

All over America, Algebra II and Precalculus students are learning how to complete the square.

Completing the square is an algebraic trick we use to solve quadratic equations. Consider this one:
$x^2-x-1=0$
It can't be factored or solved on sight. What to do? We realize that $x^2-x$ looks an awful lot like $(x-\frac{1}{2})^2=x^2-x+\frac{1}{4}$ . Thus, we add $\frac{5}{4}$ to both sides of our original equation to get
$x^2-x+\frac{1}{4}=\frac{5}{4}$
$(x-\frac{1}{2})^2=\frac{5}{4}$
The rest of the solution is as easy as square rooting both sides and doing some minor simplifications.
$x-\frac{1}{2}=\pm\frac{\sqrt{5}}{2}$
$x=\frac{1\pm\sqrt{5}}{2}$
Ta-da!

Although gajillions of methods have been used to solve quadratics throughout history, completing the square is the one most commonly used in American high schools to derive important results such as the quadratic formula.

Now watch me use it to demonstrate that $4=5$ :

$-20=-20$
$16-36=25-45$
$4^2-9\times4=5^2-9\times5$
$4^2-9\times4+\frac{81}{4}=5^2-9\times5+\frac{81}{4}$
$(4-\frac{9}{2})^2=(5-\frac{9}{2})^2$
$4-\frac{9}{2}=5-\frac{9}{2}$
$4=5$

What went wrong? The first five people to let me know at tbryant@shoshanascardin.org will be mentioned in my next post.

~Todd Bryant
Mathematics Instructor

Cake on Fridays

Monday, December 20, 2010

Completing the Square

No comments:

Post a Comment

Blog Archive