Let's start off with something easy.
Factor the expression $x^2+6x+9$.
A Little More Difficult
What do we need to add to the expression $$x^2+6x+\mbox{_____}$$ to make it into a perfect square trinomial?
Visualizing Completing the Square: $x^2+6x+\mbox{_____}$
Completing the Square in General
Question: What number do I need to add to $x^2+bx$ to get a perfect square?
Answer: $\left(\frac{b}{2}\right)^2$
Example: Complete the square. $x^2+6x$
In other words...
Question: What number goes in the blank to make a perfect square? $$x^2+bx+\mbox{_____}$$ Answer: $\left(\frac{b}{2}\right)^2$
Example: Fill in the blank with a number which. $x^2+6x$
Visualizing Completing the Square: $x^2+bx+\mbox{_____}$
Examples: Complete the square.
$x^2+4x$
$x^2-5x$
$x^2+\frac{3}{2}x$
$x^2-\frac{1}{4}x$
And what is our reward for going to all this trouble to learn how to complete the square? ...
Answer: We get to solve more equations! :D
In fact we can now solve any quadratic equation! (No matter whether it's factorable or not.)
Example: Solve the quadratic equation $x^2+3x-1=0$
Another Example: Solve the quadratic equation $$3x^2+x-1=0.$$
The Moral of the Story
Completing the square works only with $x^2+bx$.
How to Solve any Quadratic Equation: $Ax^2+Bx+C=0$
Step 0: Divide out any GCFs if necessary.
Step 1: Get your equation into the form $Ax^2+Bx=-C$.
Step 2: Divide both sides by $A$ to get an expression of the form $$x^2+bx=\mbox{some number}$$ Step 3: Add $\left(\frac{b}{2}\right)^2$ to both sides $x^2+bx$ into a perfect square.
Step 4: You should then have an equation of the form $$(x+\frac{b}{2})^2=\mbox{another number}$$ which is easy to solve.
Examples: Solve each quadratic equation by completing the square. (Don't simplify the radical expression.)
$9 \xi^2-117 \xi-171=0$
$9 t^2-3 t-7=0$
$\theta^2+\frac{1}{3} \theta-\frac{1}{4}=0$
Framing a Print A square print is surrounded by a mat and then framed. The width of the square mat is twice that of the print. If the area covered by the mat is 588 in2, determine the width of the print.
